6. Trigonometric Identities and More Equations

In Exercises 43–44, express each product as a sum or difference. sin 7x cos 3x

Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate

tan 2x + sec 2x = 3

Solve each equation over the interval [0, 2π). Write solutions as exact values or to four decimal places, as appropriate.

tan² 2x -1 = 0

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.

3 csc² x/2 = 2 sec x

In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin 2x = cos x

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.

tan² θ + 4 tan θ + 2 = 0

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.

sec² θ tan θ = 2 tan θ

Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.

sin x (3 sin x - 1) = 1

Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.

8 sec² x/2 = 4

The following equations cannot be solved by algebraic methods. Use a graphing calculator to find all solutions over the interval [0, 2π). Express solutions to four decimal places.

x² + sin x - x³ - cos x = 0

In Exercises 55–58, use the given information to find the exact value of each of the following:

c. tan(α/2)

tan α = 4/3, 180° < α < 270°

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.

2 tan θ sin θ - tan θ = 0

In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). 2 cos² x + sin x - 1 = 0

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.

9 sin² θ ― 6 sin² θ = 1

In Exercises 39–46, use a half-angle formula to find the exact value of each expression. cos 22.5°