5. Inverse Trigonometric Functions and Basic Trigonometric Equations

In Exercises 1–10, use substitution to determine whether the given x-value is a solution of the equation. __ √ 2 𝝅 cos x = ------- , x = ------ 2 4

In Exercises 1–10, use substitution to determine whether the given x-value is a solution of the equation. __ √ 3 𝝅 sin x = ------- , x = ------- 2 6

In Exercises 1–10, use substitution to determine whether the given x-value is a solution of the equation. 1 2𝝅 cos x = ﹣ ------- , x = --------- 2 3

In Exercises 1–10, use substitution to determine whether the given x-value is a solution of the equation. __ √ 3 5𝝅 tan 2x = ﹣--------- , x = --------- 3 12

In Exercises 11–24, find all solutions of each equation. __ √ 3 sin x = ------- 2

In Exercises 11–24, find all solutions of each equation. tan x = 1

In Exercises 11–24, find all solutions of each equation. 1 cos x = ﹣------- 2

In Exercises 11–24, find all solutions of each equation. tan x = 0

In Exercises 11–24, find all solutions of each equation. __ 2 cos x + √ 3 = 0

In Exercises 11–24, find all solutions of each equation. 4 sin θ﹣1 = 2 sin θ

In Exercises 11–24, find all solutions of each equation. 3 sin θ + 5 = ﹣2 sin θ

Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 2 sin² x - sin x - 1 = 0

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

Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 2 sin² x = sin x + 3

Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). sin² θ - 1 = 0

Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 4 cos² x - 1 = 0

Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 9 tan² x - 3 = 0

Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). sec² x - 2 = 0

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

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

In Exercises 53–62, solve each equation on the interval [0, 2𝝅). cot x (tan x - 1) = 0

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

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

In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin² x - 2 cos x - 2 = 0

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

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

In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). 𝝅 𝝅 sin ( x + ------ ) + sin ( x - ------ ) = 1 4 4

In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). sin x = 0.8246

In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). 2 cos x = ﹣ ------ 5

In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). tan x = ﹣3

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 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). 4 tan² x - 8 tan x + 3 = 0

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

In Exercises 127–130, solve each equation on the interval [0, 2𝝅) by first rewriting the equation in terms of sines or cosines. csc² x + csc x - 2 = 0

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

In Exercises 50–53, find all solutions of each equation. 1 cos x = ﹣----- 2

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

Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). __ √ 3 cos 4x = ﹣ --------- 2

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

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

Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). 3θ sec -------- = ﹣2 2

Exercises 25–38 involve equations with multiple angles. Solve each equation on the interval [0, 2𝝅). 3θ __ cot -------- = ﹣√ 3 2

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. 2 cos 2x + 1 = 0

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 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. cos x - 5 = 3 cos x + 6

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. 2 sin² x = 3 - sin x

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. tan x sec x = 2 tan x

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. 5 cot² x - 15 = 0

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. 7 cos x = 4 - 2 sin² x

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. 3 tan² x - tan x - 2 = 0

In Exercises 121–126, solve each equation on the interval [0, 2𝝅). 10 cos² x + 3 sin x - 9 = 0

In Exercises 121–126, solve each equation on the interval [0, 2𝝅). 3 cos² x - sin x = cos² x

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. cos 2x = -1

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. tan x = 2 cos x tan x

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. __ sin 2x = √ 3 sin x

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

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

In Exercises 12–18, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. sin 2x + cos x = 0

In Exercises 12–18, 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 + cos x = 1