Double Half Angle

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Answer Key & Explanations

Double Half Angle · Grade 12 · Worksheet 3

1. Given that cos(2θ) = 2/7 and 0° < θ < 90°, find sin(θ) = ? Answer: √(5/14) or √70/14 Solution: Use the double-angle identity cos(2θ) = 1 - 2sin²(θ). Substitute the given value: 2/7 = 1 - 2sin²(θ). Rearrange to isolate sin²(θ): 2sin²(θ) = 1 - 2/7 = 7/7 - 2/7 = 5/7.
   Full step-by-step solution

   Step 1: Use the double-angle identity cos(2θ) = 1 - 2sin²(θ). Step 2: Substitute the given value: 2/7 = 1 - 2sin²(θ). Step 3: Rearrange to isolate sin²(θ): 2sin²(θ) = 1 - 2/7 = 7/7 - 2/7 = 5/7. Step 4: Divide by 2: sin²(θ) = 5/14. Step 5: Take the square root: sin(θ) = ±√(5/14). Step 6: Since 0° < θ < 90°, sin(θ) is positive, so sin(θ) = √(5/14). Step 7: Rationalize if desired: √(5/14) = √70/14. The answer is √(5/14) or √70/14.

2. An architect is designing a modern art sculpture that consists of a triangular metal frame. The frame forms an isosceles triangle with a base of 8 meters and equal sides of 6 meters each. To calculate the optimal lighting angles, the architect needs to determine the exact value of the cosine of half the vertex angle. Using trigonometric identities, find cos(θ/2) where θ is the vertex angle of this triangular frame. Answer: √(5/6) Solution: Half-angle formulas allow us to find trigonometric values for angles that are half of known angles. For cosine, the half-angle formula is cos(θ/2) = ±√((1 + cosθ)/2), where the sign depends on the quadrant of θ/2.
   Full step-by-step solution

   Half-angle formulas allow us to find trigonometric values for angles that are half of known angles. For cosine, the half-angle formula is cos(θ/2) = ±√((1 + cosθ)/2), where the sign depends on the quadrant of θ/2. In geometric applications with triangles, angles are always between 0 and π radians, so the positive root is used. These identities are particularly useful when the cosine of the full angle is easier to determine than the half-angle itself, such as when working with triangle side lengths.

3. Aroha is a geotechnical engineer analyzing the stability of a rock slope. Using laser scanning data, she determines that the angle θ between the slope face and the horizontal satisfies sin(2θ) = 40/41, with θ in the first quadrant. To calculate the factor of safety against sliding, Aroha needs the exact value of tan(θ/2). Using double-angle or half-angle formulas, what is the exact value of tan(θ/2)? Answer: 4/5 Solution: Given sin(2θ) = 40/41 and θ is in the first quadrant, so 2θ is also in the first quadrant (since θ < 90°, 2θ < 180°, and sin positive means 2θ < 180°, and since sin is large, 2θ is between 0° and 90°).
   Full step-by-step solution

   Step 1: Given sin(2θ) = 40/41 and θ is in the first quadrant, so 2θ is also in the first quadrant (since θ < 90°, 2θ < 180°, and sin positive means 2θ < 180°, and since sin is large, 2θ is between 0° and 90°). Step 2: Use the Pythagorean identity: cos²(2θ) = 1 - sin²(2θ) = 1 - (40/41)² = 1 - 1600/1681 = 81/1681. Step 3: Take square root: cos(2θ) = 9/41 (positive because 2θ is in the first quadrant). Step 4: Use the half-angle formula: tan(θ/2) = sin(2θ) / (1 + cos(2θ)). Step 5: Substitute: tan(θ/2) = (40/41) / (1 + 9/41) = (40/41) / ((41/41 + 9/41)) = (40/41) / (50/41). Step 6: Simplify: (40/41) * (41/50) = 40/50 = 4/5. The answer is 4/5.

4. sin(2θ) = 4/5 and θ is in quadrant II, find cos(4θ) = ? Answer: -7/25 Solution: We know sin(2θ) = 4/5 and θ is in quadrant II Since θ is in quadrant II, 2θ is in quadrant III or IV Use the double-angle formula for cosine: cos(4θ) = 1 - 2sin²(2θ) Substitute sin(2θ) = 4/5: cos(4θ) = 1 - 2(4/5)² Calculate (4/5)² = 16/25 Multiply by 2: 2 × 16/25 = 32/25 Subtract: 1 - 32/25 =…
   Full step-by-step solution

   Step 1: We know sin(2θ) = 4/5 and θ is in quadrant II Step 2: Since θ is in quadrant II, 2θ is in quadrant III or IV Step 3: Use the double-angle formula for cosine: cos(4θ) = 1 - 2sin²(2θ) Step 4: Substitute sin(2θ) = 4/5: cos(4θ) = 1 - 2(4/5)² Step 5: Calculate (4/5)² = 16/25 Step 6: Multiply by 2: 2 × 16/25 = 32/25 Step 7: Subtract: 1 - 32/25 = 25/25 - 32/25 = -7/25 Step 8: The answer is -7/25

5. A circular wave pattern expands outward from a point source. At time t seconds, the radius of the circular wave front is given by r(t) = 3t² + 2 meters. A sensor is placed at a fixed point 50 meters from the source. Determine the exact time when the wave front first reaches the sensor. Answer: 4 Solution: The wave front is a circle with radius r(t) = 3t² + 2 meters. The sensor is 50 meters from the source, so the wave front reaches the sensor when the radius equals 50 meters.
   Full step-by-step solution

   Step 1: Understand the problem The wave front is a circle with radius r(t) = 3t² + 2 meters. The sensor is 50 meters from the source, so the wave front reaches the sensor when the radius equals 50 meters. Step 2: Set up the equation We set r(t) = 50: 3t² + 2 = 50 Step 3: Solve for t² Subtract 2 from both sides: 3t² = 48 Divide both sides by 3: t² = 16 Step 4: Solve for t Take the positive square root (since time cannot be negative here): t = sqrt(16) t = 4 Step 5: Interpret the result The wave front first reaches the sensor at t = 4 seconds. Final answer: 4

6. An architect is designing a curved archway for a building entrance. The arch follows the path of the function f(x) = 3sin(x) + 4cos(x). To determine the optimal placement of decorative elements, she needs to find the exact value of sin(2θ) where θ is the angle at which the arch reaches its maximum height. If the maximum occurs when tan(θ) = 4/3, what is sin(2θ)? Answer: 24/25 Solution: We are given that the maximum occurs when tan(θ) = 4/3, and we need to find sin(2θ). Recall the double-angle identity for sine. sin(2θ) = 2 sin(θ) cos(θ) We know tan(θ) = sin(θ)/cos(θ) = 4/3.
   Full step-by-step solution

   We are given that the maximum occurs when tan(θ) = 4/3, and we need to find sin(2θ). Step 1: Recall the double-angle identity for sine. sin(2θ) = 2 sin(θ) cos(θ) Step 2: We know tan(θ) = sin(θ)/cos(θ) = 4/3. Let’s represent sin(θ) and cos(θ) using a right triangle. If tan(θ) = 4/3, then: opposite = 4, adjacent = 3, hypotenuse = sqrt(4^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5. So: sin(θ) = opposite/hypotenuse = 4/5 cos(θ) = adjacent/hypotenuse = 3/5 Step 3: Plug into the double-angle formula. sin(2θ) = 2 * (4/5) * (3/5) = 2 * 12/25 = 24/25. Step 4: Conclusion. Thus, sin(2θ) = 24/25. Final answer: 24/25

7. cos(2θ) = 12/13 and 0° < θ < 90°, find sin(θ) = ? Answer: √(1/26) Solution: Use the identity cos(2θ) = 1 - 2sin²θ. Substitute cos(2θ) = 12/13: 12/13 = 1 - 2sin²θ. Rearrange: 2sin²θ = 1 - 12/13 = 1/13.
   Full step-by-step solution

   Step 1: Use the identity cos(2θ) = 1 - 2sin²θ. Step 2: Substitute cos(2θ) = 12/13: 12/13 = 1 - 2sin²θ. Step 3: Rearrange: 2sin²θ = 1 - 12/13 = 1/13. Step 4: Divide by 2: sin²θ = 1/26. Step 5: Take the square root: sinθ = ±√(1/26). Step 6: Since 0° < θ < 90°, θ is in quadrant I, where sine is positive. Step 7: Therefore, sinθ = √(1/26). The answer is √(1/26).