Trigonometric Equations

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Answer Key & Explanations

Trigonometric Equations · Grade 11 · Worksheet 1

1. Solve: 3tan²(x) - 5tan(x) - 2 = 0 for x ∈ [0, 2π]. Answer: x = arctan(2), arctan(2) + π, π - arctan(1/3), 2π - arctan(1/3) Solution: Let u = tan(x). The equation becomes 3u² - 5u - 2 = 0. Factor the quadratic: (3u + 1)(u - 2) = 0.
   Full step-by-step solution

   Step 1: Let u = tan(x). The equation becomes 3u² - 5u - 2 = 0. Step 2: Factor the quadratic: (3u + 1)(u - 2) = 0. Step 3: Solve for u: u = -1/3 or u = 2. Step 4: So tan(x) = -1/3 or tan(x) = 2. Step 5: For tan(x) = 2, the reference angle is arctan(2). Since tangent is positive in quadrants I and III, solutions are x = arctan(2) and x = arctan(2) + π. Step 6: For tan(x) = -1/3, the reference angle is arctan(1/3). Since tangent is negative in quadrants II and IV, solutions are x = π - arctan(1/3) and x = 2π - arctan(1/3). Step 7: All four solutions lie in [0, 2π]. The answer is x = arctan(2), arctan(2) + π, π - arctan(1/3), 2π - arctan(1/3).

2. Aroha is a telecommunications engineer modeling signal interference between two radio towers. The combined signal strength at a receiver is given by the equation 9 cos²θ - 12 cosθ + 4 = 0, where θ is the phase difference in radians between the two signals. Find all solutions for θ in the interval [0, 2π). Answer: θ = arccos(2/3), 2π - arccos(2/3) Solution: The equation is 9 cos²θ - 12 cosθ + 4 = 0. Recognize this as a quadratic in cosθ. Factor: (3 cosθ - 2)(3 cosθ - 2) = 0, which is (3 cosθ - 2)² = 0.
   Full step-by-step solution

   Step 1: The equation is 9 cos²θ - 12 cosθ + 4 = 0. Step 2: Recognize this as a quadratic in cosθ. Factor: (3 cosθ - 2)(3 cosθ - 2) = 0, which is (3 cosθ - 2)² = 0. Step 3: Set the factor equal to zero: 3 cosθ - 2 = 0 → cosθ = 2/3. Step 4: Since 2/3 is between -1 and 1, solutions exist. Step 5: The reference angle is arccos(2/3). Cosine is positive in quadrants I and IV. Step 6: In quadrant I: θ = arccos(2/3). Step 7: In quadrant IV: θ = 2π - arccos(2/3). Step 8: Both solutions lie in [0, 2π). The answer is θ = arccos(2/3), 2π - arccos(2/3).

3. Mere is an audio engineer designing an active noise cancellation system for an industrial workspace. Two speakers generate sound waves that interfere destructively. The net sound pressure level at a specific sensor is given by the equation 12 cos²θ - 5 cosθ - 2 = 0, where θ is the phase difference between the two waves in radians, measured in the interval [0, 2π). Determine all possible phase differences θ that produce perfect silence (zero net pressure) at the sensor. Answer: θ = arccos(2/3), 2π - arccos(2/3), arccos(-1/4), 2π - arccos(-1/4) Solution: Start with the equation 12 cos²θ - 5 cosθ - 2 = 0. Factor the quadratic in cosθ. We look for two numbers that multiply to 12 × (-2) = -24 and add to -5.
   Full step-by-step solution

   Step 1: Start with the equation 12 cos²θ - 5 cosθ - 2 = 0. Step 2: Factor the quadratic in cosθ. We look for two numbers that multiply to 12 × (-2) = -24 and add to -5. These numbers are -8 and 3. Step 3: Rewrite: 12 cos²θ - 8 cosθ + 3 cosθ - 2 = 0. Step 4: Factor by grouping: 4 cosθ (3 cosθ - 2) + 1 (3 cosθ - 2) = 0. Step 5: Factor out (3 cosθ - 2): (3 cosθ - 2)(4 cosθ + 1) = 0. Step 6: Set each factor to zero: Case 1: 3 cosθ - 2 = 0 → cosθ = 2/3. Case 2: 4 cosθ + 1 = 0 → cosθ = -1/4. Step 7: Solve Case 1: cosθ = 2/3. The reference angle is arccos(2/3). Cosine is positive in quadrants I and IV, so: θ = arccos(2/3) and θ = 2π - arccos(2/3). Step 8: Solve Case 2: cosθ = -1/4. The reference angle is arccos(1/4). Cosine is negative in quadrants II and III, so: θ = π - arccos(1/4) = arccos(-1/4) and θ = π + arccos(1/4) = 2π - arccos(-1/4). Step 9: All solutions in [0, 2π) are: θ = arccos(2/3), 2π - arccos(2/3), arccos(-1/4), 2π - arccos(-1/4). The answer is θ = arccos(2/3), 2π - arccos(2/3), arccos(-1/4), 2π - arccos(-1/4).

4. 3tan²(x) - 1 = 0 for x ∈ [0, 2π] Answer: x = π/6, 5π/6, 7π/6, 11π/6 Solution: Add 1 to both sides: 3tan²(x) = 1 Divide by 3: tan²(x) = 1/3 Take the square root of both sides: tan(x) = ±1/√3 Rationalize: tan(x) = ±√3/3 The reference angle where tan(θ) = √3/3 is θ = π/6.
   Full step-by-step solution

   Step 1: Add 1 to both sides: 3tan²(x) = 1 Step 2: Divide by 3: tan²(x) = 1/3 Step 3: Take the square root of both sides: tan(x) = ±1/√3 Step 4: Rationalize: tan(x) = ±√3/3 Step 5: The reference angle where tan(θ) = √3/3 is θ = π/6. Step 6: For tan(x) = √3/3 (positive), solutions are in quadrants I and III: Quadrant I: x = π/6 Quadrant III: x = π + π/6 = 7π/6 Step 7: For tan(x) = -√3/3 (negative), solutions are in quadrants II and IV: Quadrant II: x = π - π/6 = 5π/6 Quadrant IV: x = 2π - π/6 = 11π/6 Step 8: All solutions in [0, 2π] are x = π/6, 5π/6, 7π/6, 11π/6. The answer is x = π/6, 5π/6, 7π/6, 11π/6.

5. Aroha is an aerospace engineer modeling the vibration of a satellite's solar panel. The panel's angular displacement θ (in radians, from 0 to 2π) is governed by the equation 9sin²θ - 12sinθ + 4 = 0. Find all possible angular displacements θ in the interval [0, 2π) that satisfy this equation. Answer: θ = π/6, 5π/6 Solution: The equation is 9sin²θ - 12sinθ + 4 = 0. Recognize this as a quadratic in sinθ. Check if it factors as a perfect square: (3sinθ - 2)² = 9sin²θ - 12sinθ + 4.
   Full step-by-step solution

   Step 1: The equation is 9sin²θ - 12sinθ + 4 = 0. Step 2: Recognize this as a quadratic in sinθ. Check if it factors as a perfect square: (3sinθ - 2)² = 9sin²θ - 12sinθ + 4. Yes, it matches. Step 3: So (3sinθ - 2)² = 0. Step 4: Take square root: 3sinθ - 2 = 0. Step 5: Solve: 3sinθ = 2, so sinθ = 2/3. Step 6: The reference angle is arcsin(2/3). Since 2/3 is positive, sine is positive in quadrants I and II. Step 7: In quadrant I: θ = arcsin(2/3). Step 8: In quadrant II: θ = π - arcsin(2/3). Step 9: Both solutions are in [0, 2π). The final answer is θ = arcsin(2/3) and θ = π - arcsin(2/3).

6. Solve: 3tan²(x) - 5tan(x) + 2 = 0 for x ∈ [0, 2π] Answer: x = π/4, 5π/4, arctan(2/3), π + arctan(2/3) Solution: Let u = tan(x). The equation becomes 3u² - 5u + 2 = 0. Factor the quadratic: (3u - 2)(u - 1) = 0.
   Full step-by-step solution

   Step 1: Let u = tan(x). The equation becomes 3u² - 5u + 2 = 0. Step 2: Factor the quadratic: (3u - 2)(u - 1) = 0. Step 3: Solve for u: 3u - 2 = 0 gives u = 2/3; u - 1 = 0 gives u = 1. Step 4: So tan(x) = 1 or tan(x) = 2/3. Step 5: For tan(x) = 1 in [0, 2π]: The reference angle is π/4. Tangent is positive in quadrants I and III, so x = π/4 and x = π + π/4 = 5π/4. Step 6: For tan(x) = 2/3 in [0, 2π]: Let α = arctan(2/3) (principal value in quadrant I). Tangent is positive in quadrants I and III, so x = α and x = π + α. Step 7: The solutions are x = π/4, 5π/4, arctan(2/3), π + arctan(2/3).

7. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). The angle θ is located at the origin, with its terminal side passing through point (6,8). Find the exact value of cos(2θ) using trigonometric identities. Answer: -7/25 Solution: Find the hypotenuse of the triangle using the Pythagorean theorem Hypotenuse = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 sinθ = opposite/hypotenuse = 8/10 = 4/5 cosθ = adjacent/hypotenuse = 6/10 = 3/5 cos(2θ) = cos²θ - sin²θ cos(2θ) = (3/5)² - (4/5)² = 9/25 - 16/25 = -7/25 The answer is…
   Full step-by-step solution

   Step 1: Find the hypotenuse of the triangle using the Pythagorean theorem Hypotenuse = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 Step 2: Find sinθ and cosθ sinθ = opposite/hypotenuse = 8/10 = 4/5 cosθ = adjacent/hypotenuse = 6/10 = 3/5 Step 3: Apply the double-angle identity for cosine cos(2θ) = cos²θ - sin²θ Step 4: Substitute the values cos(2θ) = (3/5)² - (4/5)² = 9/25 - 16/25 = -7/25 The answer is -7/25.

8. sin(2x) - cos(x) = 0 for x ∈ [0, 2π] Answer: π/6, π/2, 5π/6, 3π/2 Solution: Use the double-angle identity sin(2x) = 2sin(x)cos(x) The equation becomes: 2sin(x)cos(x) - cos(x) = 0 cos(x)(2sin(x) - 1) = 0 Case 1: cos(x) = 0 Case 2: 2sin(x) - 1 = 0 → sin(x) = 1/2 Solve cos(x) = 0 in [0, 2π] x = π/2, 3π/2 Solve sin(x) = 1/2 in [0, 2π] x = π/6, 5π/6 x = π/6, π/2, 5π/6, 3π/2…
   Full step-by-step solution

   Step 1: Use the double-angle identity sin(2x) = 2sin(x)cos(x) The equation becomes: 2sin(x)cos(x) - cos(x) = 0 Step 2: Factor out cos(x) cos(x)(2sin(x) - 1) = 0 Step 3: Set each factor equal to zero Case 1: cos(x) = 0 Case 2: 2sin(x) - 1 = 0 → sin(x) = 1/2 Step 4: Solve cos(x) = 0 in [0, 2π] x = π/2, 3π/2 Step 5: Solve sin(x) = 1/2 in [0, 2π] x = π/6, 5π/6 Step 6: Combine all solutions x = π/6, π/2, 5π/6, 3π/2 The answer is π/6, π/2, 5π/6, 3π/2.