Trigonometry Regents

(x-3)/(x-3) =

1

Determine the number of radians that the minute hand of a clock travels through if it has a length of 6 inches and its tip travels a distance of 14 inches

14 = θ 6 θ = 7/3 2.3 radians

Successive Differences in Polynomials

1st difference is constant - linear function 2nd difference is constant - quadratic function 3rd difference is constant - cubic function If the difference table is not all variables, you can determine equation defining the relationship -enter tables into lists (x's into L1, y's into L2) (stat, edit, may need to clear lists first) -choose regression equation based on constant differences (linreg, cubic reg)

(2x+18)/(x²-81)

2(x+9)/(x+9)(x-9) (x+9) cancel 2/(x-9)

2x³-4x²+2/2x-2

2x³-4x²+2 2x-2(x²)=2x³-2x² (2x³-4x²+2)-(2x³-2x²)= -2x²+0x 2x-2(-x)=-2x²+2x (-2x²+2x+2)-(2x²+2x)=-2x+2 2x-2(-1)=-2x+2 (-2x+2)-(-2x+2) = 0 2x³-4x²+2/2x-2 = x²-x-1

Special triangles

30-60-90 triangle: side length √(3)/2, 1/2, 1 45-45-90 triangle: √(2)/2, √(2)/2, 1 In general with any triangle: the smallest side of a triangle is across from the smallest angle, the largest side of a triangle is across from the largest angle The 30 angle always has the √(3)/2 next to it

Reference angle of theta = 145

45

Solving system of equations

A group of equations that you deal with at the same time. You might do so graphically (graphing the equations on the same set of axes) or algebraically The solution to a system of equations is a point that makes both equations true (point of intersection) Algebraically you can solve linear equations with substitution or elimination

Standard position

An angle drawn so the initial side is on the positive side of he x-axis

Quadrantal Angles

Angles that terminate on an axis

Determine the end behavior of P(x) = 4x³-2x+7

As x -> ∞, P(x) ->∞ As x-> -∞, P(x) -> -∞

Simplifying radicals

Break down the number into 2 factors where one factor is a perfect square/cube (dependent on whether it's a cube root or square root), then simplify the perfect square/cube -4√x⁷ = -4 * √x⁶ * √x = -4 * x³ * √x = -4x³√x

Solving systems of equations with circles x²+y²+8x-14y+13=0

Completing the square Group "like" variables together Find both + (1/2 of b, squared) and add to both sides Factor

Using trig functions to find a missing angle of a right triangle when given two sides

Determine which trig function will be used based on angle you're looking for and two given sets Set up the equation Solve for the missing angle by using the inverse trig function on your calculator

P

Distance from vertex to focus OR distance from vertex to directrix

It is given that (x-1) is a factor of 4x³-12x²+3x+5, find all zeroes for the equation given below 4x³-12x²+3x+5

Divide 4x³-12x²+3x+5 by (x-1) = 4x²-8x-5 (x-1)(4x²-8x-5)=0 (x-1)(4x²+2x-10x-5)=0 (x-1)2x(2x+1)-5(2x+1)=0 (x-1)(2x+1)(2x-5)=0 x=1, x=-0.5, x=2.5

To simplify algebraic fractions

Factor and cancel

To multiply rational expressions

Factor and then cancel like factors (up/down, diagonal)

Conjugate pair/difference of perfect squares

Finding a conjugate pair (a pair of numbers that are opposites) x²-81 = (x-9)(x+9) 4x²-(y²/121) = (2x+(y/11))(2x-(y/11))

Adding or subtracting fractions

Finding the LCD (Least common denominator) Multiply on top and bottom by terms needed to make a LCD Add/subtract numerations (leave denominator) Simplify (if possible)

GCF

Finding the greatest common factor in an expression and factoring it out 6x+15 = 3(2x+5)

Parabola

Graph of a quadratic function y=ax²+bx+c

Changing parabola equations from general to standad form

Group like variables Add (completed square) to both sides Factor and move variables to opposite sides

Multiplicities

How many times a particular number is a zero for a polynomial (x-3)(x+4)²=0 -4 would have a multiplicity of 2

How will we write the equation of a parabola given the focus and directrix

Identify vertex Find p value plug in values to the equation

Factoring by grouping

If no GCF exists for the whole expression, split in half Factor out GCF for each binomial (parentheses must match) Factor out matching binomial x³+2x²+9x+18 x²(x+2)+9(x+2) (x²+9)(x+2)

Solving a radical equation containing only one radical

Isolate the radical Square both sides of the equation Solve the resulting equation Check your solution

To divide rational expressions

Keep change flip

1/(x+2) + 1/(x-2) = 4/(x²-4)

LCD: (x+2)(x-2) 1/(x+2), multiply both sides by LCD 1(x-2) 1/(x-2), multiply both sides by LCD 1(x+2) 4/(x²-4), multiply both sides by LCD 4 x-2 + x+2 = 4 2x=4 x=2 x=2 is an extraneous solution, so there is no solution

5/x + 2x/(x+2)

LCD: x(x+2) 5/x - multiply both sides by (x+2) 5(x+2)/x(x+2) 2x/(x+2) - multiply both by x 2x*x/x(x+2) (5x+10)/x(x+2) + 2x²/x(x+2) (2x²+5x+10)/x(x+2)

3/x = 8/(x-2)

LCD: x(x-2) 3/x multiply both sides by LCD 3(x-2) 8/(x-2) multiply both sides by LCD 8x 3(x-2) = 8x 3x-6=8x -6=5x x=-6/5

Solving Rational Equations

Look at all of the denominators when you are determining the LCD. you may have to factor the denominators in order to determine what the LCD for the equation will be Once you determine the LCD of all the fractions in an equation, multiply numerators by the entire LCD Keep in mind to eliminate extraneous solutions

Solving polynomial equation by factoring

Make sure equation equal zeros Factor (if necessary and possible) User zero product rule to set each factor equal to zero and solve

Solving a radical equation that contains more than one radical

Make sure there's a radical on each side before you square both sides

6/(3-√5)

Multiply both sides by 3+√5 (18+6√5)+4

2/√5

Multiply both sides by √5 2√5/5

To go from radians to degrees

Multiply by 180/pi

To go from degrees to radians

Multiply by pi/180

2x-9y=15 -5x+8y=-23

Multiply top equation by 8 Multiply bottom equation by 9 (16x-72y=120) (-45x+72y=-209) -29x=-87 x=3 2(3)-9y=15 6-9y=15 -9y=9 y=-1 (3,-1)

Factoring using AC method

Multiplying a and c values together Make a list of all pairs that would multiply up to ac value Choose the pair that adds up to the b value Replace the middle term with the pair you just chose Factor by grouping

Reference angles

Must Share same terminal side as given angle touch the x-axisbe acute R.A. in quadrant 1 = theta R.A. in quadrant 2 = 180 - theta R.A. in quadrant 3 = theta - 180 R.A. in quadrant 4 = 360 - theta

If the degree of a polynomial is 4, will we see the graph cross the x-axis 4 imes?

No, some of the roots could be imaginary

Elimination

Pick 1 variable to cancel If needed, multiply equations by values to help cancel variables Add equations and solve Solve for other variable

All Star Trig Class tells you which functions are positive

Quad. I - All are positive Quad. II - Sin is positive (+csc) Quad. III - Tan is positive (+cot) Quad. IV - Cos is positive (+sec)

Determine the ordered pair that lies on the unit circle for each angle theta = 315

R.A = 360-315 = 45 Quadrant IV 45,45,90 angle (√(2)/2, -√(2)/2

Dividing Polynomials

Reverse Tabular Method Create table # of rows- determine what kind of functions divisor is # of columns - divide high power terms on number and denominator to determine what kind of function it is place numerator on diagonal results place divisor on right hand side work backwards -OR- Long division Multiply first digit of the quotient by the divisor and then subtract the result Keep in mind: 1. When multiplying polynomials, use distributive property 2. When subtracting polynomials, add the opposite (or distribute the minus)

Secant

Sec θ = 1/Cos θ

How do we find values of x that need to be excluded when working with a rational expression

Set denominator equal to zero

In right triangle ABC, m<C = 90, a=12 and c=19. Find m<A to the neartest tenth of a degree

SinA = 12/19 Arcsin(SinA) = Arcsin(12/19) M<A = 39.166 39.2

Substitution

Solve for one variable to one equation (if needed) Substitute that equation into the other equation (in place of the variable you solved for) Solve resulting equation Solve for other variable

√(x-2) = 5

Square both sides x - 2 = 25 x = 27

Solving systems of 3 equations with 3 variable

Start by applying the elimination method to two equations in order to cancel out one of the variables Then take a different pair of equations and use elimination to cancel the same variable (this will result in having 2 equations with 2 variables) Then use the elimination or substitution method

Multiplying Polynomials

Tabular method (Creating table and multiplying each value in box) -OR- foil/distributive method (Put each factor in parentheses and start to multiply each factor by the second parentheses)

Complex numbers

The combination of real and imaginary numbers, expressed in a+bi form, where a is a real number and bi is an imaginary numbers. Complex numbers have real life applications, especially those including vectors. Vectors are how you can graph complex numbers, by using the x-axis for the real component and the y-axis for the imaginary component In order to perform operations with complex numbers, treat them as binomials. be sure to simplify powers of i and only combine like terms

Parabola (2nd definition, trig definition)

The locus (set) of all points that are equidistant from a fixed line called the directrix and a fixed point called the focus (Focus always inside parabola)

What does the degree of a polynomial tell you about the function

The number of solutions/x-intercepts

(h, k)

Vertex

If the focus of a parabola is given as (0,2), and the directrix is given as the x-axis, create an equation for the parabola

Vertex - (0,1) opens up x², +p (x-0)² = 4(1)(y-1) x² = 4(y-1)

Writing equations using real and complex factors

When given factors, work backwards, and make sure with complex numbers that the conjugate is also a factor

What does it mean for a fraction to be undefined

When the denominator is equal to zero

Remainder left in polynomial

With the remainder left over after dividing, set remainder over dividend and add to result

Rationalizing Radicals

You can't leave a radical in the denominator For a denominator that is a monomial - multiply by the smallest number needed to make the denominator a perfect square/cube For a denominator that is a binomial - you will have to multiply by the conjugate of the denominator (same binomial except opposite sign between the terms)

Cos θ

adjacent / hypotenuse

Cot θ

adjacent / opposite

Coterminal angles

angles that share the same terminal side, coterminal angles have measures that differ by multiples of 360

Sec θ

hypotenuse / adjacent

Csc θ

hypotenuse / opposite

Systems of equations can have

multiple or no solutions

Sin θ

opposite / adjacent

Sin θ

opposite / hypotenuse

You can use trig functions when looking for the missing side of a right triangle but only have the value of one side with an angle

sin(30) = x/20 20(sin(30)) = x x=10

Cos θ = Sin θ = (x,y) =

the x coordinate the y coordinate (cos θ, sin θ)

Quadratic formula

x = (-b ± √b²-4ac)/2a

x = -1, 3i

x = -1, x = 3i, x = -3i (conjugate) x+1=0, x-3i=0, x+3i=0 (x+1)(x-3i)(x+3i)=0 (x+1)(x²-9i²)=0 (x+1)(x²+9)=0 (x³+x²+9x+9)=0

(x-1)(x³+x²+x+1)

x*x³ = x⁴ x*x² = x³ x*x = x² x*1 = x -1*x³ = -x³ -1*x² = -x² -1*x = -x -1*1 = -1 combine like terms x⁴ -1

(7-x)/(x-3) Find values of x that must be excluded to prevent division by zero (creating an undefined expression)

x-3=0 x=3

When the multiplicity of a zero is even When the multiplicity of a zero is odd

x-int is tangent to x-axis x-int will cross x-axis

x²+6x+11=0

x= (-6 ± √-8)/2 x = -6 ± 2i√2 x = -3 ± i√2

Axis of symmetry

x=-b/2a

x²+y²+8x-14y+13=0

x²+8x+y²-14y=-13 x²+8x+16+y²-14y+49=-13+16+49 (x+4)(x+4)+(y-7)(y-7)=52 (x+4)²+(y-7)²=52 center - (-4,7) radius - 2√13

Direction of parabola

x², positive p value - parabola opens up x², negative p value - parabola opens down y², positive p value - parabola opens right y², negative p value - parabola opens left

y+x=5 y=x²-6x+9

x²-6x+9+x=5 x²-5x+4=0 (x-4)(x-1)=0 x=4, x=1 y=4²-6(4)+9 y=1 y=1²-6(1)+9 y=1-6+9 y=4 (1,4) , (4,1)

Sum of perfect cubes

x³+a³ = (x+a)(x²-ax+a²) + + - + x³-a³ = (x-a)(x²+ax+a²) - - + +

(xⁿ + yⁿ)(xⁿ-yⁿ)

xⁿ * xⁿ = xⁿ xⁿ * -yⁿ = -xⁿyⁿ yⁿ * xⁿ = xⁿyⁿ yⁿ * -yⁿ = -y²ⁿ combine like terms x²ⁿ - y²ⁿ

y²-2y+12x-35=0

y²-2y+1+12x-35=1 (y-1)(y-1)+12x-35=1 (y-1)²=-12x+36 (y-1)²=-12(x-3) Vertex - (3,1) Focus - (0,1) Directrix - x=6

In a circle whose radius equals 2, find the number of radians in the central angle if the arc intercept is 6

θ = 6/2, θ=3, 3 radians

The radian measure of a central angle of a circle, θ, is defined as the ratio of the length of the arc the angle subtends, s, divided by the radius of the circle, r

θ = s/r, s=θr

√(4x-7) - √(3x+9) = 0

√(4x-7) = √(3x+9) square both sides 4x-7 = 3x+9 x-7 = 9 x = 16

i

√-1

Vertex

(axis of symmetry, f(axis of symmetry))

(x²-x-2)/3 * 21/(x²-4)

(x+1)(x-2)/3 * (3*7)/(x+2)(x-2) (x-2) cancels, 3 cancels 7(x+1)/(x+2)

(x²-100)/(10-x) ÷ (2x+10)/6x

(x+10)(x-10)/10-x ÷ 2(x+10)/6x KCF (x+10)(x-10)/10-x * (2*3x)/2(x+10) Factor out (x+10), 2, (x-10) (turns to -1) -3x

x²+5=0

(x+i√5)(x-i√5)=0

x³-8

(x-2)(x²+2x+2²)

The equation of a parabola in standard form

(x-h)² = ±4p(y-k) or (y-k)² = ±4p(x-h)

(x²+5+6)(x²-3x-4)=0

(x²+2x+3x+6)(x²+x-4x-4)=0 x(x+2)+3(x+2)*x(x+1)-4(x+1)=0 (x+3)(x+2)(x-4)(x+1)=0 x=-3, x=-2, x=4, x=-1

(x-3)/(3-x) =

-1

(Cos θ)² + (Sin θ)² =

1

(x+3)/(3+x) =

1

(x+3)/(x+3) =

1