Review Exercises
Solving Trigonometric Equations with Identities
For the following exercises, find all solutions exactly that exist on the interval [0,2π).
cos2x=14
tanxsinx+sin(−x)=0
1−2tan(ω)=tan2(ω)
For the following exercises, use basic identities to simplify the expression.
sin3x+cos2xsinx
For the following exercises, determine if the given identities are equivalent.
tan3xcsc2xcot2xcosxsinx=1
Sum and Difference Identities
For the following exercises, find the exact value.
cos(25π12)
cos(83°)cos(23°)+sin(83°)sin(23°)
For the following exercises, prove the identity.
cos(3x)−cos3x=−cosxsin2x−sinxsin(2x)
For the following exercise, simplify the expression.
For the following exercises, find the exact value.
cos(sin−1(0)−cos−1(12))
Double-Angle, Half-Angle, and Reduction Formulas
For the following exercises, find the exact value.
Find sin(2θ),cos(2θ), and tan(2θ) given cosθ=−13 and θ is in the interval [π2,π].
sin(7π8)
For the following exercises, use Figure 1 to find the desired quantities.
sin(2β),cos(2β),tan(2β),sin(2α),cos(2α),and tan(2α)
For the following exercises, prove the identity.
2cos(2x)sin(2x)=cotx−tanx
For the following exercises, rewrite the expression with no powers.
cos2xsin4(2x)
Sum-to-Product and Product-to-Sum Formulas
For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.
cos(π3)sin(π4)
2cos(π5)cos(π3)
For the following exercises, evaluate the sum by using a product formula. Write the exact answer.
cos(5π12)+cos(7π12)
For the following exercises, change the functions from a product to a sum or a sum to a product.
cos(7x)cos(12x)
cos(6x)+cos(5x)
Solving Trigonometric Equations
For the following exercises, find all exact solutions on the interval [0,2π).
2sin(2x)+√2=0
For the following exercises, find all exact solutions on the interval [0,2π).
cos2x−cosx−1=0
cosx−5sin(2x)=0
For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval [0,2π). Round to four decimal places.
√3cot2x+cotx=1
For the following exercises, graph each side of the equation to find the approximate solutions on the interval [0,2π).
20cos2x+21cosx+1=0