5. Inverse Trigonometric Functions and Basic Trigonometric Equations

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

√2 cos 2θ = -1

In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). 7 sin² x - 1 = 0

Solve each equation for exact solutions over the interval [0, 2π).

―2 sin² x = 3 sin x + 1

Find all solutions of each equation. tan x = 0

Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°).

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cos x = 1/2

Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). sin 2x = √3 / 2

Find all solutions of each equation. 4 sin θ﹣1 = 2 sin θ

Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°).                     

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sin x = ―1/2

Solve each equation for exact solutions over the interval [0, 2π).

2 cot x + 1 = ―1

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.

cos θ/2 = 1

In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. sin 2x + sin x = 0

In Exercises 53–62, solve each equation on the interval [0, 2𝝅). tan² x cos x = tan² x

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

2sin θ ―1 = csc θ

Use the unit circle shown here to solve each simple trigonometric equation. If the variable is x, then solve over [0, 2π). If the variable is θ, then solve over [0°, 360°).                     

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sin θ = ―√2/2

Solve each equation for exact solutions over the interval [0, 2π).

(cot x―1) (√3 cot x + 1) = 0