Trigonometry Review

Problem 1

In the following triangle |CE|=|DE| and |BD|=4|CE|.  If the area of the trapezoid DEFG is 15, what is the area of the triangle ABC?

Problem 2

Simplify the functions f(x)=sin((tan^(-1))(x)) and g(x)=tan((sec^(-1))(x)).

Problem 3

Given that cos(72°)=(((Sqrt[5])-1)/4), find cos(144°), sin(36°), and sin(66°).

Problem 4

If θ is an angle in Quadrant III, simplify the expression

({{({{(((sin^3)(2θ))/(((sec^2)(θ)-1)(Sqrt[1-(cos^2)(θ)])))}})}})

- that is, rewrite it as trigonometric functions of θ on a single line (no denominators).

Problem 5

Rewrite 12sin(2x)-5cos(2x) in the form A sin(2x+ϕ), where A and ϕ are real numbers.

Problem 6

Write cos(3x) and cos(4x) as  formulas in cos(x).

Problem 7

If θ=(tan^(-1))((4/5)), find and simplify (sin^2)(2θ)+4cos(θ+(π/4)).

Problem 8

Solve the equation 6(cos^2)(x)-5sin(x)-2=0 over the real numbers.

Problem 9

Solve the equation sin(3x)+sin(11x)=0 over the real numbers.

Problem 10

Find 4 formulas for the graph below (of the forms A sin(B x + C)+D, and A cos(B x+C)+D, where one sine/cosine formula has A>0 and the other two has A<0).

Plot[ 3 Sin[2x+3Pi/2]+1, {x,-2Pi, 2Pi}, Ticks-> {Range[-2Pi, 2Pi, Pi/4], Range[-10,10]}, AxesStyle->Black]

Problem 11

Find formulas for the graph below of the form A sec(B x+C)+D and A csc(B x+C)+D, where A>0.

Plot[ 5-3 Sec[ 2(x-Pi/8)], {x,-Pi,Pi}, Ticks-> {Pi/8(Range[40]-20), 2Range[20]-10}, PlotRange-> {-5,15},AspectRatio-> .4,AxesOrigin-> {0,0}, Exclusions-> Cos[2(x-Pi/8)]==0, ExclusionsStyle-> Directive[ Red, Dashed], AxesStyle->Black]

Problem 12

Simplify the formula (((cos(x)+sin(x)))^4)-(((sin(x)-cos(x)))^4) , writing your answer with no powers on the functions or multiplications between functions.

Problem 13

A triangle ABC has side lengths a=2, b=5, c=6.  Find exact values for the angles in the triangle and approximate them to 3 decimal places in degree measure.

Problem 14

Suppose a triangle ABC has a=3, b=2, and B=40°.  Find the remaining sides and angles, estimating them to 3 decimal places (and using degree measure).

Common Trigonometry Concepts

Interpreting the trigonometric functions in terms of the coordinates of an endpoint

Interpreting the trigonometric functions in terms of sides of a right triangle

Values of trigonometric functions at “special angles” and their counterparts in other quadrants.

Domains, ranges, and graphs of the trigonometric functions

Inverse trigonometric functions

Similar triangles and proportionate sides

Basic identities: Even/Odd, Quotient, Reciprocal, Pythagorean

Advanced identities: Double angle, half angle, sine/cosine sum and difference, sun to product, product to sum

Law of Sines, Law of Cosines

Solving trigonometric equations

Finding formulas for a given trigonometric graph