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.

sin (x/2) = √2 ― sin (x/2)

Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 2 cos² x + 3 cos x + 1 = 0

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.

2 sin 2x ― x³ + 1 = 0

In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). cos² x - cos x - 1 = 0

In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin x cos x = √ 2 / 4

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.

√2 sin 3x - 1 = 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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sin x = ―√3/2

In Exercises 53–62, solve each equation on the interval [0, 2𝝅). (2 cos x + √ 3) (2 sin x + 1) = 0

In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 2 sin² x + sin x - 2 = 0

Find all solutions to the equation.

$\(\tan\]\theta\)=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 + 1 = 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 = √3/2  

In Exercises 127–130, solve each equation on the interval [0, 2𝝅) by first rewriting the equation in terms of sines or cosines. sec² x + 3 sec x + 2 = 0

Find all solutions of each equation. 3 sin θ + 5 = ﹣2 sin θ

In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 5 cos² x - 3 = 0