Unit 2, Unit 3, Unit 4, Unit 1-MIDTERM
co-function identity
-π/2 is 90° -x is the given angle
intercept rules: how to solve for x & y-int
1)y-int: plug in 0 for x & solve for y-ints 2)x-int: set the numerator equal to 0 & solve -EX(for x-int): f(x)=(x-3)^2/[(x+3)(x-1)] x-int= (3,0) bounce
Which is irrational: 0/2 OR 2/0
2/0 bc any # divided by 0 is undefined
what reference angle does tanX=√3/3
30 degrees (or π/6) -bc tan=opposite/adjacent=(1/2)/(√3/2)=√3/3 -so the angle of -tanX=√3/3 would be 11π/6 or 5π/6 bc they both have a reference angle of π/6 & they are in quadrants where tan is negative
what reference angle does tanX=1
45 degrees (or π/4) -bc tan=opposite/adjacent=(√2/2)/(√2/2)=1
what reference angle does tanX=√3
60 degrees (or π/3) -bc tan=opposite/adjacent=(1/2)/(√3/2)=√3
how to factor cubed polynomial
Example:x^3 - 5x^2 - 4x + 20 1)separate equation into 2 sections -(x^3-5x)-(4x+20) 3)simplify each section -x^2(x-5)-4(x-5) 3)the 2 simplified section should be the same, so make that 1 term of the factored equation & combine the coefficients of each term for the 2nd term of the factored equation -(x-5)(x^2+4)-> (x-5)(x-2)(x+2)
helpful patterns to memorize for expanding a polynomial [(a+b)^n] into standard form
In standard form: 1)find the coefficient-used Pascale's triangle to find the coefficients of small #s & use nCr(on the calculator) for larger #s -the 2 outer #s of the standard form would be 1 -1 # inside(from the outer #) is "n"(the exponent of factored form) 2) "a"'s exponent decreases from "n" to 0 3)"b"'s exponent increases from 0 to "n" EX: (x+2)^4 -> [x^4]+[4x^3(2)^1][+6x^2(2)^2]+[+4x^1+(2)^3]+[(2)^4] -> x^4 + 8x^3 + 24x^2 + 32x + 16
how many equations can a trig graph have?
Multiple -EX: a curve could be both a sin or cos graph bc everything is the same but sin starts on midline & cos starts on amplitude
what do you need to do, so that an inverse graph is a function?
Restrict the domain
which congruence postulates use law of cosine?
SAS, SSS
trigonometric functions
SOH, CAH, TOA, csc equation, sec equation, & cot equation
equation to find volume of cone -what unit is the answer in?
V=1/3 π h r^2 -h=height of triangle -r=radius of circle - units^3 (u^3)
the 2 ways to measure angles of a circle
degrees(°) or radians(π) -1 π is half of a circle or 180°
Zero Product Property
if ab=0, then a=0 or b=0
when does a curve start for a sine curve(what is the y-value of the 1st point of a parent curve, which is the point you will draw the rest of the graph based on)
it always starts at the midline
how to factor sum/difference of cubes
sum:a^3+b^3=(a+b)(a^2-ab+b^2) difference:a^3-b^3=(a+b)(a^2+ab+b^2) 1)cube the og equation to find the 1st term 2)in the 2nd term a^2 should be able to be multiplied by a from 1st term to get a^3 & b^2 should be able to be multiplied by b from 1st term to get b^3 3)in 2nd term the ab is a times b from the first term and then multiply it by the neg. sign -EX: x^3-8=(x-2)(x^2+2x+4)
factorial(!)
the product of the natural numbers less than & equal to the number that we're finding the factorial of -EX: 4!=4⋅3⋅2⋅1=24
other ways to say x-int
the root or the zero
end behavior
what happens to a function as your x-coordinates approach infinity -are the y values going up or getting more negative
from which point do you start graphing 1) sine curve 2) cosine curve (after this 1st point you keep going 1 amplitude up or 1 amplitude down)
y = asinb(x − c) + d & y= acosb(x − c) + d 1) (c, d) (starts on midline) 2) (c, d+a) (start on a peak/valley)
what is the Horizontal asymptote(HA) of every proper fraction
y=0 -bc when you have a large denominator, it keeps getting closer to 0
what is tan(60degrees) or tan(pi/3)
√3
what is tan(30degrees) or tan(π/6)
√3/3
how to find the coefficient infront of the angle for a cos or sin graph (the "b" in y = asinb(x − c) + d or y = acosb(x − c) + d)
b=2π/period (plug in the period & solve for b) -bc a period=2π/b
3 methods of dividing polynomials -when you're dividing a standard equation by a term of the factored equation
box method, long division, & synthetic division
Leading coefficient
coefficient of the leading term
how to find the important points(the points you will graph) of a cos/sin graph
divide the period by 4 & those are your x-values so find the y-values that accompany that
odd vs even degree
even: the end behaviors are the same(both sides of the parabola would go up or down) odd: the end behaviors are different(one ending would go up while other side goes down & vice versa)
difference betweeen solving for the angle of sin & arcsin (aka sin^-1) equation?
for sin ur solving for X in sin(X)which is the the input(x-value) while for sin^-1 ur solving for the the (y-value) output arcsin(√3/2)=X -for sin the variable in parentheses is the angle, while for sin^-1 the # in parentheses is the sin of the reference angle -this^ relationship applies to cos, sin, & tan with their inverse cos^-1, sin^-1, & tan^-1
what are the possible answers for the reference angle of arccos & arcsec
from 0 to π (the top half of the unit circle) -Q1 & Q2
what are the possible answers for the reference angle of arctan & arcot
from π/2 to -π/2 (aka 3π/2) (the left side of the unit circle) -Q1 & Q4
cotθ graph
-in the tanθ graph, (0,0) is the 1st parent point, but for cotθ graph, line 0 is the 1st parent asymptote
inverse
"reverses" another function: if the function(f) has an input x & gives a result of y, then its inverse function(g) is where y gives the result x, and vice versa -often inverse & reciprocal are the same, but not for trig. functions (EX: reciprocal of sin is 1/sin or csc, while inverse of sin is sin^-1 or arcsin)
how to find the range of a cscθ graph
(-∞, valley] u [peak,∞) -(opposite of the sin function)
when to use box method, synthetic division, or long division?
-box method-anytime -long division-works anytime but is harder than box method -synthetic-easiest & fastest method, but it only works when the divisor has a visible root [EX: (4x^3-2x+3)/(x+1)]-the root of the divisor is clearly 1
odd degree function -w/ a negative coefficient
-odd exponent -EX: y=-2x^3. End Behavior: As x goes to positive infinity, f(x) goes to negative infinity. As x goes to negative infinity, f(x) goes to positive infinity
odd degree function -w/ a positive coefficient
-odd exponent -EX: y=2x^3. End Behavior: As x goes to positive infinity, f(x) goes to positive infinity. As x goes to negative infinity, f(x) goes to negative infinity
bounce
-that point has a multiplicity of 2 -a vertex point resulting from a quadratic function
point of inflection
-that point has a multiplicity of 3 -a point resulting from a cubic function when the end behavior changes
how to solve for all the possible roots of a polynomial w/ integers(mod 1.9)
1) find all the possible roots of an equation using p/q -p-all the factors of the last coefficient -q-all the factors of the 1st coefficient -divide every p value by the 1st q value, then all p values by the 2nd q value & so until you've used all q values 2) use synthetic division & try all the p/q values as a factor until one of them has no remainder then it works(this takes a while) OR plug p/q values into the equation you're factoring & if f(x)=0 (no remainder), then that p/q value works & then do synthetic division(takes less time) -most roots are near the origin, so start testing the smaller p/q values, then work your way up (approx. 7 and up are unlikely) 3)If the equation resulted from dividing the og polynomial by the p/q factor is not a quadratic(has leading exponent of 2) then continue using synthetic division on that resulting equation & retry all the p/q values again to see which works this time(may be the smae p/q value or may change)(skip this if the resulting equation from step 2 IS a quadratic) 4)after achieving a quadratic equation (from using synthetic division) now you can find the roots of equation equation as you normally would(factoring/quadratic formula/completing the square)
1) how to find restricted domain of a tan graph 2) how to find restricted range of a tan graph
1) the x-value of the asymptotes surrounding the 1st point of a parent curve (the phase shift is the x-value of the 1st point of a parent curve) -same answer as sin except use PARENTHESES instead of hard brackets bc we're using asymptotes 2)the y-value of the asymptotes surround the 1st point of a parent curve-always (-∞,∞)
when does 1)sinθ=0 2)cosθ=0
1)0 & π (& the coterminals: 2π...) 2)π/2 & 3π/2 (& the coterminals: 5π/2...)
1) how to find restricted domain of a sin graph 2) how to find restricted range of a sin graph
1)[the x-value of the peak, the x-value of the valley] that surround the 1st point of a parent curve (the phase shift is the x-value of the 1st point of a parent curve) 2)[the y-value of the peak, the y-value of the valley] that surround the 1st point of a parent curve
1) how to find restricted domain of a cos graph 2) how to find restricted range of a cos graph
1)[the x-value of the the 1st point of a parent curve(the phase shift), the next point that's not a midline point(would be a peak if the 1st point of a parent curve was a valley & vice versa)] 2)[the y-value of the peak, the y-value of the valley] that surrounds the 1st point of a parent curve
box method: what if in the inner boxes, the # w/o any constant doesn't add to the last term in the standard form equation
1)add an additional number in the top right corner(the remainder), so the inner box # equals the standard form # 2)divide the remainder^ by the given divisor form and add it behind the rest of the quotient that you found -you don't need to simplify the division if you can't
how to graph a sin or cos function
1)draw midline 2) find the 1st point of the parent curve -sin curves start at the midline -cos curves start at the amplitude(if "a" is pos then starts an amplitude above midline, but if "a" is neg then it starts an amplitude below midline) 3) find & graph all the important point
How to find the equation for tan or cot graph
1)draw the asymptotes 2)draw the midline(the y line where all the inflection points are) 3)find amplitudes(the y-value distance between the inflection points & right before the line seems like it's going to go infinitely towards the asymptote) example on 4.10 edpuzzle, 19:37 4)find "b" value/the coefficient infront of the angle(b=π/b; different from sin/cos/csc/sec) 5)find phase shifts
How to find the equation for csc or sec graph
1)draw the sin graph for csc & the cos graph for sec -the vertex of the parabolas are the peaks & valleys of the sin/cos graph -the midline is between the y-value of the parabola's vertex -the x-value midline points are in between the x-values of the porabola's vertex(aka the new valleys & peaks) 2)write the sin or cos equation for the sin or cos graph you drew, but replace "sin" w/ "csc" & "cos" w/ "sec"
how to factor a quartic equation w/ the exponents: ax^4+ bx^2 + c
1)factor like you normally would for a quadratic equation(ax^2+bx+c ->(ax+b)(cx+d), but just make the 2 x values into 2 x^2 values & you may be able to factor that equation more -EX: a^4−7a^2+6=0 -> (a^2-6)(a^2-1)=0 -> (a^2-6)(a^-1)=0 2)OR use the factor by grouping method
how to find the midline from a cos graph & sin graph
1)find the y-value distance between a peak & valley point 2)divide that distance by 2(this is the amplitude) 3)substract that distance from the peak y-value point(or add that to a valley) & that is your midline
how to find the equation for the asymptote equation of any trig. graph(remember that only sec, csc, cot, & tan have asymptotes) algebraically
1)find when the trig. function is undefined(when the denominator is 0) -EX: secX is always undefined when cosX=0(bc secX=1/cos=1/0), which is always at π/2 & 3π/2 2) make the whole angle(w/ coefficients & shift)& equal that to when the trig. function is undefined (this will find one asymptote) -EX: given y=-3sec(2x-π)+1 make 2x-3=π/2 (or equal 3π/2... both work) x=3π/4 (this is one asymptote) 3) a. for sec & csc: add nperiod/2 to the asymptote you found b. for cot: nperiod to the asymptote you found -EX: x=3π/4 + π/2n
how to solve for the angle of an inverse trig. ? -EX: sin^-1(√3/2)
1)find which quadrant the angle will be in based on if it's pos. or neg. (disregard that the function is an inverse & just treat it as a normal trig. function, but limit the possible quadrants) -EX: for sin^-1(√3/2) & √3/2 is positive, so it's in Q1 or Q2 but any sin^-1 can only be in Q1 or Q4, so it's in Q1 2) find the angle in the unit circle that gives the trig. function outcome -EX: in Q1 the angle π/6 is where sin=√3/2
1)when is this helpful: nCr⋅ a^(n-r)⋅ b^r vs. 2)when is this helpful: the pattern (1.Pascal's triangle(shows all the coefficients, 2. "a" decreases from "n" to 0 3. "b" increases from 0 to "n")
1)finding a certain term in the expansion of a polynomial -this takes longer per term, but would save time for not needing to expand entire expression -EX: when asked "find the 4th term in the expansion of: (4x+3)^7" 2)expanding an entire expression -doesn't take long to expand entire expression -EX: when asked "Use Pascal's Triangle to expand the expression: (2x-3)^5"
how to graph cscθ
1)graph sinθ 2)draw dotted lines at every points (from the graph of sinθ) that are on the midline -they're going to be asymptotes for the graph of cscθ 3)graph cscθ: draw the vertex of a parabola at each peak & valley, then extend the parabola's asymptotes along the dotted lines
synthetic division
1)have the root of the divisor on the left side & all the coefficients of the standard form equation in order of descending exponents on the right side(even if the stand form storm doesn't include an x bring the # down) 2)bring down 1st(leading) coefficient & multiply by left side # 3)^write product under 2nd coefficient & add them 4)repeat until you get to the end & the last # is the remainder 5)divide the leading term's x^ exponent of the og standard form by the og divisor's constant -ex: 5x^4 was leading term of divisor expression & x was the 1st term of the divisor, so do x^4/x= x^3 6)write the newly found coefficients along w/ the x^exponent(the exponents will go down to 0) and add the remainder divided by the divisor from the given as the answer -ex: 2x^3+x^2+3x+5+23/(x-3)
Given: ax2+bx+c = 0 (they may ask you instead to find the equation based off sum &product of roots) 1) what is the sum of its roots 2)what is product of its roots
1)sum of its roots = -b/a 2)product of its roots = c/a -EX: a quadratic equation's roots have a sum of 7/6 & a product of -1/2, so the quadratic equation would be 6x^2-7x-3
difference between finding cotθ graph & tanθ graph
1)tanθ's 1st parent point is on the midline, while cotθ's 1st parent "point" is the amplitude 2) the important point above the midline for tanθ is on the right of the midline important point, but on cotθ is on the left of the midline important point 3) the important point below the midline for tanθ is on the left of the midline important point, but on cotθ is on the right of the midline important point -for cotθ the x-ints & asymptotes are switched from the tanθ graph, but everything else is the same -bc cotθ=cosθ/sinθ, while tanθ=sinθ/cosθ -so where there used to be an x-int for tanθ, replace that w/ a asymptote & vice versa
when graphing, what is the 1)x-value 2)y-value of a cosine curve
1)the given angle 2)the x-coordinate from unit circle(the cos of the given angle)
1) how to find restricted domain of an inverse trig function 2) how to find restricted range of an inverse trig function
1)the restricted range of the noninverted function 2)the restricted domain of the noninverted function -bc when you graph the inverse of a function, you switch the x & y values
how to find phase shifts for 1) tan graphs 2) cot graphs
1)the x-value of any inflection point(aka midpoint) -bc a normal tan graph w/ no P.S. has a midpoint at (0, y) 2)the x-value of any asymptote -bc a normal tan graph w/ no P.S. has an asymptote at x=0
Pascal's Triangle -what is another way to find the coefficients of the standard form after expanding a polynomial [(a+b)^n]
a pattern for finding the coefficients of the terms of a binomial expansion(the standard form) -(memorize the coefficients highlighted in blue up to the nonstandard polynomial w/ an exponent of 4) -or to find the coefficients you could use nCr by putting nCr into calculator & plugging in the exponent from the polynomial for n & the term number minus 1 for r (don't do this for smaller exponents bc you can just memorize those ones from the Pascals triangle instead of wasting time typing them into the calculator)
quotient
a result obtained by dividing one quantity by another
how to factor difference of squares
a^2-b^2=(a+b)(a-b) -do the square root of the 2 terms from the OG equation & plug into (a+b)(a-b) -EX:4x^2-9=(2x+3)(2x-3)
what is sin^-1(2) equal to? what is sin(2) equal to ?
all NA bc 2 is not on the unit circle; all trig functions(sin, csc, sin^-1, cos, sec, cos^-1) are in between 1 and -1 -the trig. functions where it's possible for the answer to not be in between 1 & -1 are the variations of tan: tan, tan^-1, cot & cot^-1
polynomial
an expression where there's at least 1 term that consists of a constant(a #) multiplied by at least 1 variable(a letter) raised to a nonnegative integral power(a whole #, positive exponent) -an exponential function is not a polynomial bc the exponent is a variable(3^x) (variable is a letter for input, while constant is a number)
even degree function -w/ a negative coefficient
as your x-coordinates approach infinity the y values are going down -even exponent -EX: y=-3x^4. End Behavior: As x goes to positive infinity, f(x) goes to negative infinity. As x goes to negative infinity, f(x) goes to negatiev infinity(the parabola points downwards)
even degree function -w/ a positive coefficient
as your x-coordinates approach infinity the y values are going up -even exponent -EX: y=3x^4. End Behavior: As x goes to positive infinity, f(x) goes to positive infinity. As x goes to negative infinity, f(x) goes to positive infinity(the parabola points upwards)
when is cotX undefined
at 0 & π -bc undefined means the denominator is 0, so: cotX=cos/sin=cos/0=(find when sin, aka y-value, is 0) 0 & π
when does sinX=0
at 0 & π -use this to find when cscX & cotX is undefined bc they have cos as a denominator
when does cosX=0
at π/2 & 3π/2 -use this to find when secX & tanX is undefined bc they have cos as a denominator
when is secX undefined
at π/2 & 3π/2 -bc undefined means the denominator is 0, so: secX=1/cos=1/0=(find when cos, aka x-value, is 0) π/2 & 3π/2
what are the possible answers for the reference angle of arcsin & arccsc
from π/2 to -π/2 (aka 3π/2) (the left side of the unit circle) -Q1 & Q4 -EX: sin^-1(2) is impossible bc sin^-1(2) -EX: -π/4 is the reference angle for sin^-1, but 7π/4 is not bc 7π/4 requires you to go through the other sections of the unit circle that are no just between π/2 to -π/2, so it's not possible
Rational Zero Theorem
if a polynomial has integer coefficients, then each of its rational zeros has the form p/q -p-factors of the last coefficient -q-factors of the 1st coefficient
how dyk if a graph shows a function
if the same x-value has only 1 or no y-value -(the vertical line test: if you draw a vertical line & it passes through more than 1 point, then it's not a function)
when does a curve start for a cosine curve(what is the y-value of the 1st point of a parent curve, which is the point you will draw the rest of the graph based on)
it always starts at a peak(if the coefficient in the very front of the right side of the equation is pos.) or valley(if the coefficient in the very front of the right side of the equation is neg.)
what is the inverse of a trig. function?
it's the trig. function to the power of negative one -not csc or sec, but sin^-1 & cos^-1 bc: csc/sec is the reciprocal of sin/cos (cscX∙sinX=1), while sin^-1/cos^-1 is the inverse (sin^-1∙sinX= 0.0087≠1)
how to find any term on a polynomial (when changing (a+b)^n to standard form)
nCr⋅ a^(n-r)⋅ b^r -r-term number minus 1 -a, b, n-from the og factored equation: (a+b)^n -nCr=(n!)/[r!(n-r)!] (just put nCr into calculator, u don't have to memorize nCr equation)("nCr" is aka "combination")
for imaginary roots we always have a...
negative & positive answer(the conjugate of the imaginary root is also a root of the equation) -EX: x=4i find the equation. <-this means that -4i is also a root -EX:3+2i <- this means 3-2i is also a root
difference between finding cotθ graph & -tanθ graph
negative tanθ means that the point are reflected so the above midline points are now on the left of the midline point, which is just like cotθ, but the phase shift indicated on the cotθ gives the 1st asymptote, while the P.S. on the -tanθ graph gives the 1st midline point, so there's just a shift difference between the 2
reciprocal
one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2
box method
put the standard form of the divisor on one side of the outer table 1) fill in the top left corner box w/ the leading term from the standard form equation 2) find a # that when multiplied by the divisor term gives the term in the top left box & write the # on the outside above it 3) fill in the bottom left box by multiplying outside the table #s (the one you found & the given #) 4)to find the box 2nd to the top left: the bottom left most corner(solved for in step 3) & the box 2nd to the top left should add to the 2nd term in the given standard form 4)continue to solve for the rest of the inner boxes & by doing that you will find the other eqxpression, which is above the table -to find some inner boxes: the inner boxes w/ constants to the same exponent should add to the standard equation
how to graph secθ
same as how you graph a cscθ except you 1st graph cosθ instead of sinθ
what should you do when an answer(such as answer from solving for an x-int) has a square root of a negative number? -EX: a=√-1 & a=√-6
take the negative out of the root & turn it to i (imaginary number -EX: i & i√6
what does it show on a graph when tanx=undefined (so if 0 was the denominator)
that is on the vertical asymptote
multiplicity
the # of times a function goes through a specific x-int
how to find the period(per) from any trig. graph
the distance between any 2 points that are equally spaced out throughout the graph & that have the same y-value (but remember that when using 2 midlines points' x-value, those 2 points should have a valley & a peak in between) -a period should start at a midline point, include one valley & one peak, then end on the next midline point(for csc & sec when you draw the sin or cos graph when finding the equation & the same applies) -or to find the distance of a period it could start at 1 peak(or 1 valley) & end at the next peak(or valley) (this is OK to calculate the distance of a period, but is not officially the definition of a period) -EX: x-value distance between adjacent midpoint & midpoint, OR valley & valley, OR peak & peak
what is arcsin?
the inverse of sin (sin^-1) -arctan=sin^-1, arcsos=cos^-1, arccsc=csc^-1, arcsec=sec^-1
least degree 1)how to find least degree from a graph 2)how to find least degree from equation
the smallest degree that a function can have (it's the exponent # of the leading term) 1)least degree=the number of turns + 1 -EX: 5x, the least degree is 1 bc there are 0 turns 2)the highest exponent out of the equation -if the function is even, then it will have an even # as a least degree & if the function is odd, then it will have an odd # as a least degree
leading term
the term w/ the highest exponent in a polynomial EX: g(x)=x^4−3x^3+2x^5−8x+5, leading term is 2x^5
degree
the total amount of exponents in factored form & the highest exponent in standard form -indicates the type of polynomial -EX: (x^2+7)(x^2-2) has a degree of 4 & shows it's a quartic function -least degree is different than degree bc in the EX above the least degree would be 2? -EX: 2x^2+6x^3+7x+5 has a degree of 3
how many possible answers are there for the angle of a trig. function(EX: what is sinx= -1/2)
there are an infinite amount of answers bc there are an infinite amount of coterminals
how many possible answers are there for the angle of an inverse trig. function(EX: what is sin^-1 x= -1/2)
there's only 1 answer -bc we restricted the domain & range, so that there's only 1 y-value for each x-value, which caused there to only be 1 x-value when solving for it
when do you use Rational Zero Theorem?
to find the factors of an equation that's not factorable & we're not given a factor/zero -EX: you can't factor x^3+3x^2-11x+12 by grouping bc the 2 separated groups don't have matching factors, so use this method -EX: i don't think there's any method to factor quartic(other than quartic equation w/ the exponents 4, 2, & 1) or larger polynomials w/o this method (4x^4-x^3-3x^2+9x-10)
define turns 1) how to find number of turns given a graph 2)how to find number of turns given an equation
when a line changes directions on a graph (doesn't matter if it's a bounce from a squared term or an inflection from a cubic function their turns would both count as 1) 1)the # of times the line changed/ when there's a curve -EX(look in photo): there are 3 turns 2)highest exponent of the equation minus 1 -EX x^4: 4-1=3, so 3 turns
conjugate
where you change the sign in the middle of two terms -EX: 3+x & 3−x are conjugates.
Quadratic Formula
x = -b ± √(b² - 4ac)/2a when ax^2+bx+c=0
how to find when the asymptotes of a cscθ or a secθ graph occur
x= the 1st, smallest positive asymptote +/- 1/2period -1/2 a period is the distance between each midline point -also all the asymptotes are when cscθ=undefined, so when the denominator is 0 -0 is a positive -you can use any asymptote, but it's easier to just use the 1st, smallest positive asymptote
how to find the equation for the midline points algebraically (you won't be required to do this)
x=the 1st positive midline point(make y=0 in the given trig. equation & solve for x) + periodn -EX y=2tan4(x+π)-1: 0=2tan4(x+π)-1, x=, x=smthn
where is the 1)amplitude 2)vertical shift(where our midline is) 3)helps determine the period in a trig function 4)phase/horizontal shift 5) domain of a trig function graph(sin, cos, tan, sec, csc, cot) 6) range of a trig function graph(sin, cos, tan, sec, csc, cot)
y = asinb(x − c) + d 1)coefficient infront of trig function(a) -for sin: after the 1st point of a curve(on the midline) the next important point(right) is 1 amplitude up, but if "a" is negative then the next important point is 1 asymptote down the midline -for cos: a parent curve usually starts from above the midline, but if "a" is negative then you start curve below midline -EX: y=2cos would start the parent curve at a y-value of 2, but y=-2cos would start the parent curve at y=-2 2)the # being added(d) -y=d 3)coefficient infront of the angle(b) -usually a period is 2π, then the coefficient infront of the angle would make the period 2π/b -there should never be a coefficient infront of x in the angle (x − c), so take that coefficient out of the parentheses 4)horizontal(phase shift; moves graph left of right)(c) -x MINUS c means you're going right c -this is where your parent curve starts(starts at c instead of the y-axis) 5) for csc, cot, sec, tan: R, x≠asymptote for sin, cos:(-∞,∞) -means all real numbers except for when x= asymptotes 6)for csc, cot, sec, tan: (-∞,∞) for sin, cos: [y-value of the valleys, y-valley of the peaks]
how to find the phase shift(P.S.) from a 1) cos graph 2) sin graph
y = asinb(x − c) + d or y = acosb(x − c) + d 1) -if you make "a" pos: the x-value of ANY valley(they all work) -if you make "a" neg: the x-value of ANY peak 2) -if you make "a" pos: the x-value of ANY point on the midline where the peak is directly after that point -if you make "a" neg: the x-value of ANY point on the midline where the valley is directly after that point
how to graph tanθ(in 7 steps)
y = atanb(x − c) + d 1)state the amplitude(a), period(π/b), phase shift(c) 2)plot the 1st parent point -on the midline, 1 phase shift 3)plot points on the midline: the next point is 1 period away from the previous one 4)draw asymptotes: in the middle of the midline points 5)draw the main points above the midline: in between a midline point & the closest asymptote, the y-value goes up from the midline by 1 amplitude(a) 6)draw the important points below the midline: in between a midline point & asymptote, the y-value goes down from the midline by 1 amplitude(a) 7) connect points & use asymptotes!
how to find # of pieces of functions to sketch on a graph
# of functions = 1 more piece than the # of vertical asymptotes
Sum & Difference Formulas for Csc of a given angle -what about sec & cot?
(it's the Sum & Difference Formulas for Sin of a given angle OVER 1) 1)sin(u+v)=1/[(sinu)(cosv)+(cosv)(sinv)] -u & v should add to the given angle 2)sin(u-v)=1/[(sinu)(cosv)-(cosv)(sinv)] -Sum & Difference Formulas for csc, sec, & cot are just Sum & Difference Formulas for sin, cos, & tan but over 1
how to make a coordinate of a unit circle
(length, height) -length is x value & height is y value
Basic Identities/Pythagorean Identities
(they're all version of the Pythagorean trig. identities. EX: when you divide 1st equation by sin^2(θ) you get 3rd equation & when you divide 1st equation by cos^2(θ) you get 2nd equation)
what happens if you have a value that's both the x-int & a vertical asymptote?
It becomes neither, it creates a hole(it would never occur if you remembered to simplify your fraction before solving VA & x-int) -EX: the x-ints of an equation is (1,0) & (-1,0) & the VAs are x=-4 & x=-1, so cross out -(they must've forgot to simplify the og fraction)
How to find unknown sides & angles of AAS or ASA triangles if we only know a few pieces of information about them?
If ABC is a triangle with sides a, b, &c, then: a/sinA = b/sinB = c/sinC or it can be written as: sinA/a = sinB/b = sinC/c -that is the law of sines^
what is a weird case of a triangle that uses law of sines? -What additional step is needed for solving for missing parts of this triangle?
SSA -1)you'll need to first see if the given angle is obtuse or acute 2)figure out whether it has 0, 1, or 2 possible triangles 3)if it as 0 possible triangles the answer is NA, 1 possible triangle solve for it as you normally would, 2 possible triangle take the 1st degree you solved for when normally solving for 1 possible triangle then subtract that from 180 & then use law of sines to solve for new missing parts w/ the new extra angle in addition to the given angle & sides
what does open circle mean?
it's when it's an undefined #(a # that when plugged in as x will make the denominator 0)
do all improper fractions have slant asymptotes?
no, improper fractions have horizontal asymptotes: 1)if there are no constants 2)OR if the highest DEGREE(don't care abt coefficients) in the numerator & denominator are the same(equal)
a reference angle is always...
positive, acute, & formed w/ the x-axis -while coterminal or given angle could be neg. or pos.
slant asymptote (aka oblique asymptote)
replaces the position of the horizontal asymptote when the "horizontal asymptote" has a constant -EX: y=x-1
vertical asymptote rule (how to find VA))
set the denominator equal to 0 & solve -bc doing that^ would make the graph undefined
Pythagorean identity
sin²θ + cos²θ = 1 -same as Pythagorean theorem except we known that the hypotenuse is always 1 bc it's a unit circle(remember that cosine θ=x value on unit circle & sineθ=y value on unit circle)`
how to find SA(slant asymptote) equation of an improper fraction
-EX: f(x)=(x^2+2x+5)/(x+3) 1) divide the numerator by the denominator -> f(x)= x-1 - (8/x+3) 2) remove the remainder & you've got you SA -> f(x)=x-1 -bc the proper remainder (8/x+3) keeps getting smaller as x-values increase, so it doesn't really matter as x values go to infinity & negative infinity
half angle formulas 1)for sin of a given angle 2)for cos of a given angle 3)for tan of a given angle
-a is the given angle times 2(so when you divide a by 2 you get the given angle) -for sin & cos equation you have to decide whether it's positive or negative, which depends on the ORIGINAL given angle(remember sin=y value, so if y value is negative then it's negative & cos=x value, so if x value is negative then it's negative) -for the tan equation both equations get the same answer
what is the degree(& give reference angle) 1)π/6 2)π/4 3)π/3 4)2π/3 5)3π/4 6)5π/6 7)7π/6 8)5π/4 9)4π/3 10)5π/3 11)7π/4 12)11π/6
-any radian w/ the denominator 6 is 30°, w/the denominator 3 is 60°, w/ the denominator 4 is 45° 1)30° 2)45° 3)60° 4)120°(60°) 5)135°(45°) 6)150°(30°) 7)210°(30°) 8)225°(45°) 9)240°(60°) 10)300°(60°) 11)315°(45°) 12)330°(30°)
What are the law of cosines equations if ABC is a triangle w/ sides a, b, & c -when dyk which equation to use?
-for SAS use the equation w/ the side across from the given angle as the single term on the left -for SSS you can use any equation, but she recommends using the equation w/ the biggest side as the single term on the left
how to find what shape is formed when a nonright triangle is rapidly spinning around y-axis? -what is that shape? -how do you find the volume of that shape?
-frustum of a cone(a cone w/ the top half cut off) w/ a hole in the shape of a cone inside -V = (1/3)π h (R^2 + r^2 + Rr) - ((1/3)π h r^2)
What does 1)SOH 2)CAH 3)TOA stand for?
-hypotenuse is opposite of 90 degree
Double angle formulas
-θ is the given angle divided by 2(so when θ is multiplied by 2 it will equal the given angle) -sin(2θ) & cos(2θ) could also be solved for using sum & difference formulas sin(θ+θ) & cos(θ+θ) -(under cos(2θ) all the equations are interchangeable, so memorize them all bc they'll be helpful in verifying identities)
when a triangle is SSA, what does it mean if they have 0 triangle, 1 triangle, 2 triangles?
0 triangles) the answer for the missing angle or side is no solution(bc the shape is not a triangle, so you can't you law of sines 1 triangle)solve for the missing angle/side with law of sines just like for normal triangles that follow congruence postulates 2 triangles)take the angle from the 1st triangle that you solved for & subtract from 180, then add that new angle to a new triangle w/ the og given values & solve for new values
what are all the unit circle angles
0°(or 360°) 30° 45° 60° 90° 120° 135° 150° 180° 210° 225° 240° 270° 300° 315° 330° -the closest angle to the y & x-axis is the axis +/- 30° -the angle between the x-axis is the axis +/- 45° OR the closest angle +/- 30°
what are the proportions between the side lengths of a triangle w/ a 30, 60, & 90 degree (use the constant a) 1)hypotenuse 2)long leg 3) short leg
1)2a 2)a√3 3)a
SSA, ASA, SSS, SAS, AAS: 1)which of these triangles use law of sines 2)which use law of cosines?
1)Law of Sines: SSA, ASA, & AAS -for SSA you need to solve for unknown values for 0, 1, or 2 possible triangles 2)Law of Cosines: SSS & SAS
How to find the general solution of possible angles given a trig function
1)Simplify the equation if needed -factor -if the variables are different(cos^2x + sinx-1=0) use basic identities to make them the same variable(like turn cos^2x into 1-sin^x) -EX: if ur given 0=cosX(csc^2X-2), then you'll need to separate the factors (cosX=0 & csc^2X-2=0 & solve X separately) 2)Find what possible quadrant the angle could be in -(EX: if it's sin(x)=-1 it must be in quadrant 2 or 3 bc those are the quadrants with negative sine values) 3)draw the sides in the quadrant(s) to find the angle -EX: sin(x)=√3/2, so the opposite side of x is √3/2, so the angle x must be 60 degrees 4)add 2πn(n is the number of terms) -EX(final answer of sin(x)=√3/2): x=√3/2 + 2πn -bc there are an infinite amount of conterminal angles so you can keep adding 2πn -if the the 2 possible angles from 2 quadrants are directly across from eachother, forming 1 line, then the general equation can be the smaller angle plus πn(EX: 4.3 Solving Trig Equations, 14:54) -if the 4 possible angles from 4 quadrants are 45 degrees, then the general equation can be the smallest angle plus (π/2)n(EX: 4.3 Solving Trig Equations, 20:13)
what are the proportions between the side lengths of a triangle w/ a 45, 45 & 90 degree (use the constant a) 1)hypotenuse 2)long leg 3) short leg
1)a√2 2)a 3)a -the "short" leg & the "long" leg would be the same length, so one's not shorter than the other
Sum & Difference Formulas for Cos of a given angle
1)cos(A+B)=(cosA)(cosB)-(sinA)(sinB) 2)cos(A-B)=(cosA)(cosB)+(sinA)(sinB)
what are the reciprocal of 1)sine (sin) -what does this equal? 2)cosine (cos) -what does this equal? 3)tangent (tan) -what does this equal?
1)cosecant (csc) -1/sinθ= hypotenuse/opposite 2) secant (sec) -1/cosθ= hypotenuse/adjacent 3) cotangent(cot) -1/tan = adjacent/opposite
end behavior rules(how to find end behavior)
1)end behavior is f(x) approaching 0 is when the the denominator is greater than the numerator -bc the # is getting REALLY big or REALLY small so what is happening is that the numerator would be WAY less than the denominator making it closer and closer to 0 2)when the denominator & numerator have equal degrees(exponents) the end behavior is the numerator's leading coefficient over the denominator's leading coefficient -EX: (2x-3)/(4x+1) end behavior is f(x)->∞, x->1/2 & as f(x)->-∞, x->1/2 3)when the numerator's highest degree is greater than the denominator's highest degree the end behavior will follow the normal end behavior of the biggest terms divided -EX (x^2)/x: dividing the variables degrees you get x^1 (aka just x) & the normal end behavior for that is f(x)->∞, x->∞ & as f(x)->-∞, x->-∞ -EX (x^3 - 2x)/(x+3): dividing the highest degrees(x^3/x) you get x^2, so the end behavior is is f(x)->-∞, x->∞ & as f(x)->-∞, x->∞
How to solve for the hole
1)have the thing that you canceled from both sides of the fraction equal 0 & solve for x -EX: we simplified (x+2)/[(x+2)(x-2) into 1/(x-2), so we canceled out (x+2) from both sides of the og equation, so we do 0=x+2, then x=-2 2)THEN plug in that x value we found into the simplified equation, which gives the y-value of the point -EX: simplified equation is 1/(x-2) & we plug in x=-2, which is 1/-4, so y=-1/4 and our answer is (-2,-1/4)
in a right triangle where is the hypotenuse, short leg, & long leg?
1)hypotenuse= long slanted leg, across from the 90 degree angle 2)short leg =across from the 30 degree angle(or the smallest angle) 3)long leg=across from the 60 degree angle(or the largest angle of the 2 legs' angles)
how to find reference angle when the angle is in the 1)1st quadrant(0° to 90°) 2)2nd quadrant(90° to 180°) 3)3rd quadrant(180° to 270°) 4)4th quadrant(270° to 360°)
1)reference angle = angle 2)reference angle = 180° - angle 3)reference angle = angle - 180° 4)reference angle = 360° - angle
Sum & Difference Formulas for Sin of a given angle
1)sin(A+B)=(sinA)(cosB)+(cosB)(sinA) -A & B are angles from the unit circle that should add to the given angle 2)sin(A-B)=(sinA)(cosB)-(cosA)(sinB)
Sum & Difference Formulas for Tan of a given angle
1)tan(A+B)=(tanA+tanB)/(1-tanAtanB) 2)tan(A-B)=(tanA-tanB)/(1+tanAtanB)
when graphing, what are the 1)x-values 2)y-values of a sin curve
1)the given angle(θ) 2)the y-coordinate from unit circle(the sin of the given angle)
for simplifying fractions w/ fractions on either sides when should you 1)inverse the overall denominator then multiply by overall numerator 2)multiply both sides of the fraction by values to cancel denominators of both sides
1)when the denominator(s) of all the values of the overall denominator is the same value -EX: Mod 2 Quiz #3 2)when the denominator(s) of all the values of the overall denominator are different -EX: Mod 2 Quiz #4
guidelines for verifying identities
1)work w/ 1 side at a time(the side which is complex, so you need to simplify & if both sides look complex, then choose the side that looks like there's a good step to help you solve) 2)look for chances to factor, add fractions, square binomials, or create 1 term denominators 3)Look for chances to use basic identities 4)If nothing else works turn everything into sines & cosines
what coordinate does 1) cosA 2) sinA 3) tanA equal in a unit circle
1)x coordinate 2)y coordinate 3)y coordinate/x coordinate (bc sine/cosine) -makes sense bc in a unit circle the hypotenuse is 1, so when sin=opposite/hypotenuse & cos=adjacent/hypotenuse the denominator(hypotenuse) is always 1, so the value is just the y value(opposite) or x value(adjacent)
1)how to solve for missing angle in the domain(0, 2π] when tan(x)=0 (yes zero 0, not theta θ) 2)what about when tan(3x)=0
1)x= 1 & -1 -bc tan=sin/cos=y/x, so you look for whenever y=0 which would be at y=1 & -1 2)x=1/3 & -1/3 -bc the angle 3x=1 & -1, so x= 1/3 & -1/3
what is(remember to imagine a unit circle's special right triangle): 1)cos(30) 2)sin(30) 3)cos(60) 4)sin(60) 5)cos(45) 6)sin(45)
1)√3/2 2)1/2 3)1/2 4)√3/2 5)√2/2 6)√2/2
rational number
A number that can be written as a fraction
isosceles triangle
A triangle that has 2 equal sides & therefore the 2 sides opposite of them are equal
which congruence postulates use law of sines?
AAS, ASA, SSA
how to find area of a SAS triangle
Area = 1/2 bc sinA -remember the b,c, & A variables are arbitrary, so b & c are the given sides & A is the angle in-between -if it's not an SAS triangle you can just find use sine or cosine to make sure there are 2 known sides and 1 known angle then use this formula
how to find area of a SSS triangle(Herron's Formula) -what's another way to find SSS triangle area?
Area = √[s(s − a)(s − b)(s − c)] -solve for s: Semi-Perimeter = (a + b + c)/2 -or you could use law of cosines to find 1 angle then use SAS area formula, but that would take longer
end behavior
The behavior of the graph(what the y-values do) as x approaches positive infinity or negative infinity (horizontal asymptote) -EX: f(x)->∞, x->0 & as f(x)->-∞, x->0
equation to find volume of a cylinder -what unit is the answer in?
V=π h r^2 -units^3(u^3) (any 3d shape is in some unit^3)
what is the reference angle for tanX=√3
X= 60° or π/3 bc remember tan=sin/cos, so when sine is divided by cosine it should equal √3, so the only options in the unit circle would be sin=√3/2 & cos=1/2 since (√3/2)/(1/2)=(√3/2)x(2) = √3
frustum of a cone -equation to solve for its volume
a cone w/ the top half cut off -V = (1/3)π h (R^2 + r^2 + Rr) -r is radius of top circle(bc small circle=small r) -R is radius of bottom circle(bc large circle=large R)
improper fraction
a fraction whose numerator is larger than the denominator -EX: 35/7 is improper bc it should be 5
hole
a hole is the point on the graph caused by simplifying an expression -EX: when we simplify (x+2)/[(x+2)(x-2) into 1/(x-2), the graph will have a point representing the canceling of the (x+2)
trapezoid
a quadrilateral with one pair of parallel sides
vertical asymptote
a vertical(up or down) line that a graph approaches but never crosses -EX: x=-1
what do you need to remember when solving for the angle of a trig function that includes half, double, triple, etc. angles?
after you got your general angle answers, you need to go back and add to each answer the thing that you were adding(πn , 2πn, π3n, or something else) until you get to 2π(or 360 degrees) (on 16:30 on Asynch 4.3 Edpuzzle)
what should you always do 1st before finding the x-ints, y-ints, VAs, & end behaviors?
always factor the dominators & numerators & cancel out any same factors -EX: (x+2)/[(x+2)(x-2) has x-int at (-2,0), but it's wrong bc you need to 1st simplify the equation into 1/(x-2), which shows that there are no x-ints
coterminal angle(define) 1)how to find coterminal angle? 2)how many coterminal angles does a given angle have?
angles with the same terminal side -in the pic: so if 120 degrees is the given angle, then -240 degrees is a coterminal angle 1)-find it by adding or substracting 360 to the given angle 2)infinite amounts bc you can keep adding or subtracting 360 degrees to the given angle
how to find area of a triangle -When should you use this equation?
area = 1/2(base)(height) -Base & height must be perpendicular to each other(the height going through the triangle must form a right angle) -Don't use this equation, use the SSS or SAS area equations bc they don't require you to waste time in solving for h
Pythagorean Theorem
a²+b²=c² where a and b represent the legs of a right triangle & c represents the hypotenuse
When the triangle is SSA & the given angle(A) is obtuse what are the # of possible triangles?
b is the side attached to A & a is the side opposite of A(A is the given angle) 1) a≤b no triangle 2) a>b 1 triangle
When the triangle is SSA & the given angle is acute what are the possible triangles?
b is the side attached to A, a is the side opposite of A, & h is the altitude(sticks out between side a & b) 1) a>b 1 triangle 2)h>a no triangle 3)h=a 1 triangle 4)h<a<b 2 triangles -1st see if a>b bc you don't need to solve for h if that's right
how to find what shape is formed when a nonlight triangle(isosceles or equilateral) is rapidly spinning around x-axis? -what is that shape? -what's the equation to find the volume of the 2 cones
connect the pointy edges sticking out w/ a circular shape(there should be 1 circular shapes) -2 cones stacked on top of each other w/their circular base touching -V=2/3 π h r^2 (basically multiplying the og cone volume equation by 2)
how to find what shape is formed when a right triangle is rapidly spinning around x-axis? -what is that shape?
connect the pointy edges sticking out w/ a circular shape(there should be 1 circular shapes) -cone
when to use half angle formulas
given angles that are not on the unit circle & cannot be subtracted/added up by unit circle angles(adding 45, 30, 60 or 90 degrees does not equal that given angle), so you must divide an angle on the unit circle by 2, which equals the given angle & use the half angle formulas
how to find the altitude(h) of a SSA triangle? -where is the altitude in a triangle?
h=b sin A (b is the side attached to the given angle & A is the given angle) -h sticks out from the angle between side b & side a
how to you know if an angle is obtuse
has a measurement greater than 90 degrees but less than 180 degrees
how to you know if an angle is acute
has a measurement less than 90 degrees
slant/ oblique asymptote
if the degree of the numerator is exactly one more than the degree of the denominator -when the "horizontal asymptote" is infinity, which means it does not just have a slope of 0 like horizontal lines, but a slope of 1
what makes a function(a(x)) over another function(b(x)) an improper fraction
if the degree(highest exponent) of a(x) is greater than or equal to the degree of b(x) -EX:(x^3-x^2)/(x^2-4x) is improper -we count exponents & not coefficients bc exponents make a much bigger difference -remember that just because an equation is improper does not make it have a slanted asymptote bc if numerator & denominator have same degree, then the have horizontal asymptote
what do critical values tell us?
if you plug in the x-values between the critical values & the inequality is true, then shade the area between the nearest 2 undefined values, which tells you where(the x-values of) one function follows the inequality
quotient identity
tan=sin/cos & cot=cos/sin -this makes sense bc in a unit circle cosine θ=x value & sineθ=y value -remember a quotient=a result obtained by dividing one quantity by another
reference angle
the acute angle formed by the terminal side IN the quadrant that the angle terminates and the x-axis -remember angles of a circle form anti-clockwise, so the terminal side is the one further in an anticlockwise clock -if a given angle is in the (positive, positive) quadrant, then the reference angle is the same as the given angle
leading coefficient
the coefficient of the leading term
one period of a sine curve
the curve/graph formed when the x-values being plotted are all from the first 360 degrees(or 2radian) -they repeat after this -to find the period, it's the x-value distance from a midline point then passes through a peak & valley, to the other midline point
amplitude
the distance from the peaks/valleys to the midline -(how high/low we go from the midline)
how to find domain
the domain should account for the vertical asymptotes & the holes' x-value(if there is no hole then don't account for it) -EX: vertical values are x=4,1 & it has a hole at (-1, -1/2) so the domain of the graph is (-∞, -1) u (-1,1) u (1, 2) u (4, ∞)
what should you do 1st before using law of sines to solve for missing side or angle?
the known values of the triangle(the parts of the triangle which are actually given as a #) should be in the order of AAS OR ASA
midline -what is for cosine & sine parent graphs
the line which the valleys & peaks always go back to -always y=0(or x-axis) for cosine & sine parent graphs
which trig functions are positive in quadrant 1, 2,3, & 4
the memory aid is A(ll) S(tudents) T(ake) C(alculus) 1) All trig functions are positive 2)only Sin & its reciprocal(csc) 3)only Tan & its reciprocal(cot) 4)only Cos & its reciprocal(sec)
leading term
the term w/ the highest exponent in a polynomial -EX: g(x)=x^4−3x^3+2x^5−8x+5, leading term is 2x^5
what do the x-values mean when found from 2 expressions that equal each other?
the x-values means the point where the 2 functions on either sides of the equation intersect on a graph
critical values
they are the solutions & undefined values(which x value will make the denominator of a fraction 0); they help form the interval notation for inequalities(test any value between these critical values by plugging them back into the inequality to see if it's still true & if it's true then that's a correct interval) -these only occur for inequalitie; Edpuzzle 2.2, 7:50)
how to solve for missing values in SSS triangle & SAS triangle
uses law of cosines TWICE then 1)for SAS: uses law of sines 2)for SSS: subtract angles from 180 to find last angle
equation to find volume of a sphere
v=4/3πr³
general domain angles
when given a trig. function with a missing angle, the general domain angles are the infinite amount of possible angles (includes coterminal angles)
restricted domain angles
when given a trig. function with a missing angle, the restricted domain angles are the finite amount of possible angles (no coterminal angles); the angles between (0, 360°]
when do you use Sum & Difference Formulas
when the angle given is not on the unit circle, but 2 angle numbers which could add or subtract to that given angle are on the unit circle(EX: cos15 = cos(45-30))
standard position
when the initial point of a vector is at the origin
zero product property
when the product of two or more factors is zero, one of these factors must equal zero (EX: if ab=0, then a=0 and/or b=0)
when is the answer no solution when trying to find the angle of a sin trig function?
when the sinX is not between -1 & 1 -same applies to cosX, cscX, & secX -EX: sinX=-4/3 is no solution
when is an x-intercept(or any other point) extraneous?
when you plug the x-value back in the given equation & it makes the denominator 0 or/and when a factor can be canceled from the denominator & numerator
a quick way to find the HA of a fraction when the numerator & denominator have same highest degree
y=(highest coefficient of the numerator)/(highest coefficient of the denominator) -EX: (x^2-16)/(x^2-6x+8) you can immediately see the HA is y=1 bc the highest coefficients are both 1 so 1/1=1