Graphing tangent and reciprocal
Take the _____ of A
Absolute value
Finding your bearings
Always find them by looking at north or south line first
In cosine, _____ equals 0 In cosine, the reference angle is the _____ of the other angle Which numbers are neg and pos?
Anything over 2 that doesn't reduce to whole number like 5pi/2 or 3pi/2 Opposite sign! For example pi/6 is sqrt3/2 but 5pi/6 is -sqrt3/2
Find the exact value of each of the remaining trig functions of theta. Cos theta=-24/25, theta is in quad 3
Draw out triangle Y is negative because theta is in quad 3
Composite functions! f(f-1(x))=x. f-1(f(x))=x
F-1(x) must be in domain of f f(x) must be in domain of f-1
Graphing tangent functions
Find consecutive asymptotes using -pi/2<bc-c<pi/2 Find midway between asymptotes Find 1/4 pt ( LA plus mdpt divided by 2) and 3/4 pt (mdpt plus RA divided by 2) Y-copies -A and A respectively
Sin ( sqrt2/2) vs sin-1(sqrt2/2)
First gives decimal second gives pi
INVERSE PROPERTIES tan(tan-1x)=x tan-1(tanx)=x
For all reals For all x in (-pi/2, pi/2)
KEEP IN MIND: If there's a negative before the function...ex: y=-2cos2x LABEL POINTS SO MR. HAYBURN CAN SEE
If reflects over x-axis
The whole even and odd function rule doesn't hold true for...
Inverses Cos-1(-sqrt3/2) equals 5pi/6 not neg
Positive cosine graph look like
Maximum at origin
In sin cos and tan, the reference angles are all
Positive
Given sin theta=0.8836
Put calc in degree and use inverse button
If the absolute value of A is smaller than one, the graph...
Shrinks
When given sin(sin-1) or sin-1(sin) use
The inverse properties
If given sin (sin -1 pi) what happens
not in domain so its undefined when its the inverse in the parenthesis and its not in domain, its UNDEFINED
If given sin -1 (sin 5pi/4) what happens
number not in domain but since inverse is outside of parenthesis, you can still do it find sin 5pi/4 then then sin -1 of that
Pythagorean identities
sin2(t) + cos2(t) = 1 tan2(t) + 1 = sec2(t) cot2(t) + 1 = csc2(t)
Y=sin-1x means ____ where
x=siny where -pi/2 is less than or equal to y which is less than or equal to pi/2 and -1 is less than or equal to x which is less than equal to 1
Sine inverse chart -1, -sqrt3/2, -sqrt2/2, -1/2, 0, 1/2, sqrt2/2, sqrt3/2, 1
-pi/2 -pi/3 -pi/4 -pi/6 0 pi/6 pi/4 pi/3 pi/2
Y=sinx has inverse on
-pi/2 lessthan or equal to x less than or equal to pi/2
Sin 0 Cos 0 Tan 0
0 1 0
Sin-1 0 Cos -1 0 Tan -1 0
0 Pi/2 0
Y=cosx has inverse on
0 is less than or equal to x less than or equal to pi
Cosθ= 0, pi/6, pi/4, pi/3, pi/2, 2pi/3, 5pi/6, pi, 7pi/6, 4pi/3, 3pi/2, 5pi/3, 11pi/6, 2pi Remember cosine pattern!
1 sqrt3/2 sqrt2/2 1/2 0 -1/2 -sqrt3/2 -1 -sqrt3/2 -1/2 0 1/2 sqrt3/2 1
Cosine inverse chart 1, sqrt3/2, sqrt2/2, 1/2, 0, -1/2, -sqrt2/2, -sqrt3/2, -1
1, pi/6, pi/4, pi/3, pi/2, 2pi/3, 3pi/4, 5pi/6, pi
Number line
1/8 1/4 1/2 3/4 1 but with pi!
For example, y=4sin(2x-2pi/3) What's A B and C?
A is 4 B is 2 and C is 2pi/3
If given tan -1 (-1)
A quick way would be to take the negative out and get the answer then throw it back in
y=cotx domain and range
All real numbers except multiples of pi All real numbers (-inf, inf)
When putting sin(0.2) in calculator,
BE IN RADIAN MODE
Finding angle of elevation, calc should be in
Be in degrees
phase shift
C/B and it's horizontal
ON TEST: sin-1(1.3)
Can't do it because it's outside domain
Even functions
Cos sec
Find two values of theta, 0 less than equal to theta less Han equal o 2pi thst satisfy he equation: cos theta=sqrt2/2
Find where cos is positive then find two angles in those quadrants
Graphing tan in calc:
Put function in Y1 and put Xmin as -pi/2 and Xmax -3pi/2 and Xscale pi/4
ASTC?
Q1 all positive Q2 sin and Csc are + rest - Q3 tan and cot are + rest - Q4 cos sec are + rest are -
The domain of the restricted function for sinx cosx and tanx are the _____ for sin-1x cos-1x and tan-1x
RANGE
REMEMBER SIN AND COS PATTERN
Sin: 0 1 0 -1 0 1 0 -1 0 1
Amplitude changes...
The minimum and maximum NOT THE PERIOD
Function domain and interval The restrictions are...
The range for y=sin-1x y=cos-1x and y=tan-1x
How to solve sin49 csc49
They're reciprocals so sin49(1/sin49) Which equals 1
How to figure out if it's a function?
Vertical line test
Trig using y r and x Sin Cos Tan Csc Sec Cot
Y/r X/r Y/x R/y R/x X/y
y=cscx domain and range
all real except multiples of pi (-infinity, -1] U [1, infinity)
y=secx domain and range
all real numbers except odd multiples of pi/2 (-inf, -1] U [1, inf)
Finding the side length of triangle, put calc in
degrees
INVERSE PROPERTIES Sin(sin-1(x))=x Sin-1(sin(x))=x
for all x in [-1, 1] for all x in [-pi/2, pi/2]
Inverse and intercepts are
not the same
Where's tangent undefined??
pi/2 3pi/2
What's 30 45 60 and 90 in radians
pi/6 pi/4 pi/3 pi/2
put calc in degree or radians: cos -1 6/7
radians
Reciprocal Identities
sinθ = 1/cscθ ; cscθ = 1/sinθ cosθ = 1/secθ ; secθ = 1/cosθ tanθ = 1/cotθ ; sin/cos cotθ = 1/tanθ cos/sin
45-45-90 triangle
x, x, x√2
30-60-90 triangle
x, x√3, 2x x is across from 30 x√3 is across from 60 2x is across from the hypotenuse
What does y=sin-1x mean
x=siny
sin 0, cos 0, tan 0
0 1 0
Sin θ= 0, pi/6, pi/4, pi/3, pi/2, 2pi/3, 5pi/6, pi, 7pi/6, 4pi/3, 3pi/2, 5pi/3, 11pi/6, 2pi
0 1/2 sqrt2/2 sqrt3/2 1 sqrt3/2 1/2 0 -1/2 - sqrt3/2 -1 -sqrt3/2 -1/2 0 Remember sin pattern!
Tanθ= 0, pi/6, pi/4, pi/3, pi/2, 2pi/3, 5pi/6, pi, 7pi/6, 4pi/3, 3pi/2, 5pi/3, 11pi/6, 2pi
0 sqrt3/3 1 sqrt3 undefined -sqrt3 -sqrt3/3 undefined sqrt3/3 sqrt3 undefined -sqrt3 -sqrt3/3 0
GRAPHING VARIATIONS OF Y=SINX STEPS
1. Identify the amplitude and period 2. Find the x-value of the five key points (3 intercepts, 1 min, 1max) 3. Find the y-values 4. Connect points in a smooth curve 5. Extend graph as desired
In graphing tangent, how do you find 1/4 point? 3/4 point?
Add left asymptote and midpoint and divide by 2...use -A for y Add midpoint and right asymptote...use A for y
Odd functions
All functions are odd except cos and sec
y=cscx domain and range
All real except multiples of pi (-inf, -1] U [1, inf)
Y=tanx domain and range
All real numbers All real except odd multiples of pi/2
Y=cos-1x means
Cosy=x where 0 which is less than or equal y which is less than or equal to pi and -1 is less than or equal to x which is less than or equal to 1
Finding distance across a lake in which would be solving for one of the sides, make sure your calc is in
Degrees
Graphing a tangent function:
Do -pi/2<bx-c<pi/2 Find midpoint between them Then find 1/4 and 3/4 point
To find inverse sin cos and tan, just
Do the unit circle and draw a triangle using 45 45 90 and 30 60 90
Domain and range of tangent function?
Domain all real except multiples of pi/2 Range is all reals
Domain and range of y=cosx
Domain is (-inf, inf) Range is [-1, 1]
Domain and range of y=sinc
Domain is (-inf, inf) Range is [-1, 1]
Domain and range of y=cotx
Domain is all real except multiples of pi Range is all real
Domain and range of y=secx
Domain is all real except odd multiples of pi/2 Range is (-inf, -1] U [1, inf)
Domain and range of y=cscx
Domain is all real numbers except multiples of pi Range is (-inf, -1] U [1, inf)
The point given below is on the terminal side of an angle theta. Find the exact value of each of the six trig functions of theta. (3, -3)
Find r R=sqrt x2+y2 Then use the ratios for example sin is y/r
INVERSE PROPERTIES cos(cos-1x)=x cos-1(cosx)=x
For all x in [-1,1] For all x in [0, pi]
What happens if a negative is in front of sin or cos?
For sin when it's positive the pattern is 0 1 0 -1 0 When negative it's 0 -1 0 1 0 and reflect across x axis
Increment is
How far each point is from one another
In sine, the fractions that have a smaller number in the numerator, The fractions that have a larger number in numerator equal In sin, any number ___ is 0
It's positive Example: 2pi/3 and 5pi/6 They have the reference angle of pi/3 and pi/6 and equal sqrt3/2 and 1/2 Negative Any number like 2pi 3pi 11pi
For problems like sin(sin-1 ...)
Just cancel them out and use x When it written w positive sin then neg sine, it's just the number This goes for ALL trig functions
Graphing trig functions in calc...
Make sure to be in RADIAN mode
A negative sin graph looked like
Maximum then 0 then minimum
Negative cosine graph look like
Minimum at origin
A positive sin graph looks like
Minimum second quadrant then goes thru 0 then maximum in first quad
Does y=sinx pass the horizontal line test?
No, so we look at domain -pi/2< or equal to x< or equal to pi/2 which PASSES the HL test
Which trig functions have amplitudes
ONLY sin and cos
Is sin an even or odd function
Odd
Inverse function
Passes the horizontal line test If point (a,b) is in f(x) then point (b,a) is in f-1(x) It is the reflection of the function about the line y=x
cos-1(cos pi/2)
Pi/2 is in the domain so it's pi/2
The whole thing with inverse and no inverse is basically seeing what you're given and basically finding the reference angle...so Q1 Q2 Q3 Q4 When it's a non acute angle
Q1, it's just itself Q2 180- theta Q3 theta - 180 Q4 360 - theta 360 - theta And if it's negative, go counterclockwise
θ 0 30 45 60 90 0 π/6 π/4 π/3 π/2 SIN COS TAN
SIN 0 1/2 sqrt2/2 sqrt3/2 1 COS 1 sqrt3/2 sqrt2/2 1/2 0 TAN 0 sqrt3/3 1 sqrt3 undefined tangent is sin/cos so you can always get tangent
Which functions do you graph the same way?
SIN AND COS
Simplifying expressions of sin -1x involving algebraic expressions cos(sin -1 x)
Set sin -1x= θ sin θ=x which is sin θ=x/1 which is o/h So set up triangle like before on graph and find a Then evaluate it at cos Should be like sqrt 1-x^2
If given one trig function then another, example, cos(tan-1 tan 5/12) you...
Set the one inside the parenthesis to theta: tan-1 5/12= θ and then rewrite it like tan θ=5/12 Well tan is o/a which is y/x so set it up lie a triangle on coordinate plane! Then once you have that you evaluate it at the trig function in front of parenthesis! Which is cosine
Cofunctions
Sin and cosine are cofunctions: sin(pi/2-x)=cosx cos(pi/2-x)=sinx Tangent and cotangent are cofunctions: tan(pi/2-x)=cotx cot(pi/2-x)=tanx Secant and cosecant are cofunctions: sec(pi/2-x)=cscx csc(pi/2-x)=secx
Diff between sin of a regular fraction like sin sqrt2/2 or sin 1/2 and sin of a pi like sin pi/4
Sin of a fraction like sin 1/2 gives you just a decimal Sin of a pi like sin pi/4 gives you a fraction like the quiz we had
cos(cos-1(pi/2))
So pi/2 is within the domain so the answer is pi/2
Sin pi/4 Cos pi/4 Tan pi/4 Sin pi/3 Cos pi/3 Sin pi/6 Cos pi/6
Sqrt2/2 Sqrt2/2 1 Sqrt3/2 1/2 1/2 Sqrt3/2
What's the starting point?
Starting point is 0 when there's no c When there's c the stating point is C/B
If the absolute value of A is larger than one, the graph...
Stretches
Which trig functions are which period
Tan and cot have the period of pi or pi/b All the rest have the period of 2pi or 2pi/b
Questions about values and inverse's?
Use ASTC ALL STUDENTS TAKE CALC
Can't graph y= Csc sec or cot so you
Use reciprocal!
Graph Csc and sec?
Use reciprocals!! For Csc use sin For sec use cos The x intercepts of sin and cos are the VERTICAL ASYMPTOTES The minimum of sin and cos are the maximums The maximum of sin and cos are the minimums SKETCH!
Where do you put vertical asymptotes at on a tangent graph?
Vertical asymptotes at pi/2 and all odd multiples of pi/2(1 3 5 7)
Y=tanx vs y=tan -1 x
With the inverse, the x and y switch Domain of regular is (-pi/2, pi/2) and range is (-inf, inf) Domain of inverse is the range of regular, (-inf, inf) the range is the domain of regular (-pi/2, pi/2) Regular has a Points at (-pi/4, -1) (0,0) and (pi/4,1) Inverse has points at -1, -pi/4 and 0,0 and 1, pi/4 Regulars two vertical asymptotes of - and + pi/2 are now horizontal asymptotes
Graphing cot?
You'd think you'd use a reciprocal but graph it like tangent! Except use 0<bx-c<pi Find midpoint between them Find 1/4 and 3/4 Points THIS TIME..use A first then -A Think that the A's are switched because tan and cot are reciprocals
When finding SinA, you have to make sure
Your calculator is in DEGREE MODEEE
Function: y=tanx Domain of restricted function Interval on which the function complete one period
[-pi/2, pi/2] (-pi/2, pi/2)
Function: y=sinx Domain of restricted function Interval on which the function complete one period
[-pi/2, pi/2] [0, 2pi]
For finding the inverse, go by these restrictions ...KEEP THESE IN MIND EACH TIME YOU DO INVERSE Sin-1x Cos-1x Tan-1x
[-pi/2, pi/2] [0, pi] (-pi/2, pi/2]
Function: y=cosx Domain of restricted function Interval on which the function complete one period
[0, pi] [0, 2pi]