Pre-Calculus Quiz 6-1 PDF

# Pre-Calculus - Quiz 6-1 ## Name: Nikolina Dimily ## Date: ## Per: ## Unit 6: Trigonometric Identities & Equations ## Quiz 6-1: Basic Trigonometric Identities/ Proving Trigonometric Identities ### Use the Reciprocal and Quotient Identities **for questions 1 and 2.** **1. Find csc x if sin x...

# Pre-Calculus - Quiz 6-1 ## Name: Nikolina Dimily ## Date: ## Per: ## Unit 6: Trigonometric Identities & Equations ## Quiz 6-1: Basic Trigonometric Identities/ Proving Trigonometric Identities ### Use the Reciprocal and Quotient Identities **for questions 1 and 2.** **1. Find csc x if sin x = 2√5 / 5** * csc x = 1/sin x * csc x = 1 / (2√5 / 5) * csc x = 5 / 2√5 * csc x = √5 / 2 **2. Find cos a if tan a = √2 / 2 and sin a = - √3 / 2** * tan a = sin a / cos a * cos a = sin a / tan a * cos a = (-√3 / 2) / (√2 / 2) * cos a = -√3 / √2 * cos a = -√6 / 2 ### Use the Pythagorean Theorem Identities **for questions 3 and 4.** **3. Find sin x if cot x = -√3 / 2 and cos x < 0.** * cot x = 1 / tan x * cot x = cos x / sin x * cos x = cot x * sin x * cos² x = cot² x * sin² x * sin² x = 1 - cos² x * sin² x = 1 - (cot² x * sin² x) * sin² x + cot² x * sin² x = 1 * sin² x *( 1 + cot² x) = 1 * sin² x = 1 / (1 + cot² x) * sin² x = 1 / (1 + (√3 / 2)²) * sin² x = 1 / (1 + 3 / 4) * sin² x = 1 / (7 / 4) * sin² x = 4 /7 * sin x = 2 / √7 * sin x = 2√7 / 7 **4. Find cot θ if cos θ = -7 / 12 and sin θ < 0.** * cot θ = cos θ / sin θ * cot θ = cos θ / √(1 - cos²θ) * cot θ = (-7 / 12) / √(1 - (-7 / 12)²) * cot θ = (-7 / 12) / √(1 - 49 / 144) * cot θ = (-7 / 12) / √(95 / 144) * cot θ = (-7 / 12) / (√95 / 12) * cot θ = -7 / √95 * cot θ = -7√95 / 95 ### Use the cofunction and even-odd identities **for questions 5 and 6.** **5. Find tan (-x) if cot (-x) = -1.84.** * tan (-x) = 1/cot(-x) * tan (-x) = 1 / -1.84 * tan (-x) = -1 / 1.84 * tan (-x) = -0.54 **6. Find cos (-π/2 + θ) if sin θ = -0.57.** * cos (-π/2 + θ) = sin θ * cos (-π/2 + θ) = -0.57 ### For questions 7-10, simplify the trigonometric expression. **7. sec x * sin² x + cos x** * sec x * sin² x + cos x * (1/cos x) * sin² x + cos x * sin² x / cos x + cos x * (sin² x + cos² x) / cos x * 1 / cos x * sec x **8. csc ß - sin ß / sin ß** * csc ß - sin ß / sin ß * (1/sin ß) - sin ß / sin ß * (1 - sin² ß) / sin ß * cos² ß / sin ß **9. sin y + sin y * cot² y / csc y** * sin y + sin y * cot² y / csc y * sin y + sin y * (cos² y / sin² y) / (1 / sin y) * sin y + cos² y / sin y * (sin² y + cos² y) / sin y * 1 / sin y * csc y **10. (sin θ + 1) * (tan θ - sec θ)** * (sin θ + 1) * (tan θ - sec θ) * (sin θ + 1) * (sin θ / cos θ - 1 / cos θ) * (sin θ + 1) * (sin θ - 1) / cos θ * (sin² θ - 1) / cos θ * (-1 * (1 - sin² θ)) / cos θ * (-1 * cos² θ) / cos θ * -cos θ ### For questions 11-12, prove the trigonometric identity. **11. 1 + sec² x / 1 + tan² x = sec² x** * 1 + sec² x / 1 + tan² x * 1 + 1/cos² x / 1 + sin² x / cos² x * (cos² x + 1) / cos² x / (cos² x + sin² x) / cos² x * (cos² x + 1) / cos² x / 1 / cos² x * (cos² x + 1) / cos² x * cos² x * cos² x + 1 * sec² x **12. sin a / 1 - cos a * cot a = csc a** * sin a / 1 - cos a * cot a * sin a / 1 - cos a * cos a / sin a * sin² a / sin a - cos a * cos a * sin² a / sin a - cos² a * sin² a / sin a - (1 - sin² a) * sin² a / sin a (1 - 1 + sin² a) * sin² a / sin² a * 1 / sin a * csc a