Trigonometric Identities and Equations

CONCEPT PREVIEW Which one is not a linear equation? A. 5x + 7 (x - 1) = -3x B. 9x² - 4x + 3 = 0 C. 7x + 8x = 13 D. 0.04x - 0.08x = 0.40

Solve each linear equation. See Examples 1–3. 7x + 8 = 1

Solve each linear equation. See Examples 1–3. 7x - 5x + 15 = x + 8

Solve each linear equation. See Examples 1–3. 6(3x - 1) = 8 - (10x - 14)

Solve each linear equation. See Examples 1–3. 2 [x - (4 + 2x) + 3] = 2x + 2

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

Solve each linear equation. See Examples 1–3. 5 4 5 — x - 2x + —— = —— 6 3 3

Solve each linear equation. See Examples 1–3. 3x - 1 x + 3 ———— + ———— = 3 4 6

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

Solve each linear equation. See Examples 1–3. 0.2x - 0.5 = 0.1x + 7

Solve each linear equation. See Examples 1–3. 0.01x + 3.1 = 2.03x - 2.96

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

Solve each linear equation. See Examples 1–3. -4 (2x - 6) + 8x = 5x + 24 + x

Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. 4(2x + 7) = 2x + 22 + 3(2x + 2)

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

Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. 2(x - 8) = 3x - 16

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

Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. -2(x + 3) = -6(x + 7)

Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. 4(x + 7) = 2(x + 12) + 2(x + 1)

Solve each quadratic equation using the zero-factor property. See Example 5. x² - 5x + 6 = 0

Solve each quadratic equation using the zero-factor property. See Example 5. 5x² - 3x - 2 = 0

Solve each quadratic equation using the zero-factor property. See Example 5. -4x² + x = -3

Solve each quadratic equation using the zero-factor property. See Example 5. x² - 100 = 0

Solve each quadratic equation using the zero-factor property. See Example 5. 4x² - 4x + 1 = 0

Solve each quadratic equation using the zero-factor property. See Example 5. 25x² + 30x + 9 = 0

Solve each quadratic equation using the square root property. See Example 6. x² = 16

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

Solve each quadratic equation using the square root property. See Example 6. x² - 27 = 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. tan x = 2 cos x tan x

Solve each quadratic equation using the square root property. See Example 6. (3x - 1)² = 12

Solve each quadratic equation using the quadratic formula. See Example 7. x² - 4x + 3 = 0

Solve each quadratic equation using the quadratic formula. See Example 7. 2x² - x - 28 = 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. __ sin 2x = √ 3 sin x

Solve each quadratic equation using the quadratic formula. See Example 7. x² - 2x - 2 = 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. 5 cos² x - 3 = 0

Solve each quadratic equation using the quadratic formula. See Example 7. x² - 6x = -7

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

Solve each quadratic equation using the quadratic formula. See Example 7. -2x² + 4x + 3 = 0

Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. 5x +2 ≤ -48

Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. -2x + 8 ≤ 16

Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. -2x - 2 ≤ 1 + x

Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. -3(x - 6) > 2x - 2

Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. 3(x + 5) + 1 ≥ 5 + 3x

Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. 2x - 5 ———— < 5 4

Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. 4x + 7 ———— ≤ 2x + 5 -3

Solve each inequality. Give the solution set using interval notation. See Example 10. -9 ≤ x + 5 ≤ 15

Solve each inequality. Give the solution set using interval notation. See Example 10. -5 < 5 + 2x < 11

Solve each inequality. Give the solution set using interval notation. See Example 10. 10 ≤ 2x + 4 ≤ 16

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

Solve each inequality. Give the solution set using interval notation. See Example 10. x + 1 -4 ≤ ———— ≤ 5 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. 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. 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