Triangle Trigonometry

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Triangle Trigonometry · Grade 11 · Worksheet 2

1. Aroha is standing at a point on a cliff edge, observing two boats on the ocean. The boats are located at points B and C, and Aroha's position is point A. The distance from Aroha to boat B is 45 meters, and the distance from Aroha to boat C is 60 meters. The angle between the lines of sight to the two boats is 52 degrees. Find the distance between the two boats, correct to the nearest meter. Answer: 48 Solution: Identify the given information in triangle ABC. Side AB = 45 m, side AC = 60 m, and included angle BAC = 52 degrees. We need side BC.
   Full step-by-step solution

   Step 1: Identify the given information in triangle ABC. Side AB = 45 m, side AC = 60 m, and included angle BAC = 52 degrees. We need side BC. Step 2: Use the Law of Cosines: BC^2 = AB^2 + AC^2 - 2(AB)(AC) * cos(angle BAC). Step 3: Substitute the values: BC^2 = 45^2 + 60^2 - 2(45)(60) * cos(52). Step 4: Calculate the squares: 45^2 = 2025, 60^2 = 3600. So BC^2 = 2025 + 3600 - 2(45)(60) * cos(52). Step 5: Sum the squares: 2025 + 3600 = 5625. Step 6: Calculate 2(45)(60) = 5400. Step 7: So BC^2 = 5625 - 5400 * cos(52). Step 8: Find cos(52) using a calculator: cos(52) = 0.61566 (approximately). Step 9: Multiply: 5400 * 0.61566 = 3324.564. Step 10: Subtract: 5625 - 3324.564 = 2300.436. Step 11: Take the square root: BC = sqrt(2300.436) = 47.96 (approximately). Step 12: Round to the nearest meter: 48 m. The distance between the two boats is 48 meters.

2. sin(π/4)cos(π/12) + cos(π/4)sin(π/12) = ? Answer: √3/2 Solution: Recognize the trigonometric identity: sin(A)cos(B) + cos(A)sin(B) = sin(A+B) Identify A = π/4 and B = π/12 Calculate A + B = π/4 + π/12 = 3π/12 + π/12 = 4π/12 = π/3 The expression simplifies to sin(π/3) sin(π/3) = √3/2 The answer is √3/2.
   Full step-by-step solution

   Step 1: Recognize the trigonometric identity: sin(A)cos(B) + cos(A)sin(B) = sin(A+B) Step 2: Identify A = π/4 and B = π/12 Step 3: Calculate A + B = π/4 + π/12 = 3π/12 + π/12 = 4π/12 = π/3 Step 4: The expression simplifies to sin(π/3) Step 5: sin(π/3) = √3/2 The answer is √3/2.

3. In triangle ABC, side a = 16, side b = 11, and angle C = 60°. Find side c. Answer: 14 Solution: Use the Law of Cosines: c² = a² + b² - 2ab cos(C) Substitute the given values: c² = 16² + 11² - 2(16)(11) cos(60°) Calculate squares: c² = 256 + 121 - 2(16)(11)(0.5) Simplify: c² = 377 - 352(0.5) = 377 - 176 Subtract: c² = 201 Take square root: c = √201 ≈ 14.177 Round to nearest whole number: c…
   Full step-by-step solution

   Step 1: Use the Law of Cosines: c² = a² + b² - 2ab cos(C) Step 2: Substitute the given values: c² = 16² + 11² - 2(16)(11) cos(60°) Step 3: Calculate squares: c² = 256 + 121 - 2(16)(11)(0.5) Step 4: Simplify: c² = 377 - 352(0.5) = 377 - 176 Step 5: Subtract: c² = 201 Step 6: Take square root: c = √201 ≈ 14.177 Step 7: Round to nearest whole number: c = 14 The answer is 14.

4. Mere is a marine biologist tracking a pod of dolphins. She observes two points where dolphins surface: Point A and Point B, which are 120 meters apart along a straight coastline. From Point A, she measures the angle to a distant buoy as 38°. From Point B, she measures the angle to the same buoy as 54°. Both angles are measured from the coastline. How far is the buoy from Point B, to the nearest meter? Answer: 74 Solution: Let the triangle have vertices A (Point A), B (Point B), and C (buoy). Side AB = 120 m. Angle at A = 38°, angle at B = 54°.
   Full step-by-step solution

   Step 1: Let the triangle have vertices A (Point A), B (Point B), and C (buoy). Side AB = 120 m. Angle at A = 38°, angle at B = 54°. Step 2: Find angle C: 180° - 38° - 54° = 88°. Step 3: Use the Law of Sines: side_a / sin(angle A) = side_b / sin(angle B) = side_c / sin(angle C). We want the distance from Point B to the buoy, which is side AC (opposite angle B). Label side AC = b (opposite angle B). Step 4: Set up proportion: b / sin(54°) = 120 / sin(88°). Step 5: Calculate sin(54°) ≈ 0.8090, sin(88°) ≈ 0.9994. Step 6: b = 120 * 0.8090 / 0.9994 ≈ 120 * 0.8095 ≈ 97.14. Step 7: That gives side AC, but we want side BC (distance from Point B to buoy). BC is opposite angle A (38°). Label BC = a. Step 8: Use Law of Sines: a / sin(38°) = 120 / sin(88°). sin(38°) ≈ 0.6157. Step 9: a = 120 * 0.6157 / 0.9994 ≈ 120 * 0.6161 ≈ 73.93. Step 10: Round to nearest meter: 74 m. The answer is 74.

5. In triangle ABC, side a = 17, side b = 22, and angle C = 72°. Find side c. Answer: sqrt(1037 - 748*cos(72°)) Solution: Substitute the given values: c^2 = 17^2 + 22^2 - 2(17)(22) cos(72°). Calculate the squares: 17^2 = 289, 22^2 = 484. Sum the squares: 289 + 484 = 773.
   Full step-by-step solution

   Step 1: Apply the Law of Cosines: c^2 = a^2 + b^2 - 2ab cos(C). Step 2: Substitute the given values: c^2 = 17^2 + 22^2 - 2(17)(22) cos(72°). Step 3: Calculate the squares: 17^2 = 289, 22^2 = 484. Step 4: Sum the squares: 289 + 484 = 773. Step 5: Calculate the product: 2(17)(22) = 748. Step 6: So c^2 = 773 - 748 cos(72°). Step 7: Take the square root: c = sqrt(773 - 748 cos(72°)). Step 8: Using a calculator, cos(72°) ≈ 0.3090, so 748 * 0.3090 ≈ 231.132, then 773 - 231.132 = 541.868, and sqrt(541.868) ≈ 23.28. The answer is sqrt(773 - 748 cos(72°)) or approximately 23.28.

6. In triangle Hana, side a = 12, side b = 20, and angle C = 120°. Find side c. Answer: sqrt(544) Solution: Use the Law of Cosines: c² = a² + b² - 2ab·cos(C). Substitute a = 12, b = 20, C = 120°: c² = 12² + 20² - 2(12)(20)·cos(120°). Calculate squares: 12² = 144, 20² = 400.
   Full step-by-step solution

   Step 1: Use the Law of Cosines: c² = a² + b² - 2ab·cos(C). Step 2: Substitute a = 12, b = 20, C = 120°: c² = 12² + 20² - 2(12)(20)·cos(120°). Step 3: Calculate squares: 12² = 144, 20² = 400. Step 4: Sum: 144 + 400 = 544. Step 5: Compute product: 2(12)(20) = 480. Step 6: cos(120°) = -1/2. Step 7: So c² = 544 - 480·(-1/2) = 544 + 240 = 784. Step 8: Take square root: c = sqrt(784) = 28. The answer is 28.

7. Mason is a surveyor mapping a triangular plot of land. He measures two sides as 27 meters and 32 meters, and the angle opposite the 27-meter side as 37°. Mason needs to determine if this information defines a unique triangle, two possible triangles, or no triangle at all. What is the nature of this triangle configuration? Answer: Two possible triangles Solution: Identify given values. Side a = 27 m (opposite angle A = 37°), side b = 32 m (opposite angle B). Use Law of Sines: a/sin(A) = b/sin(B) → 27/sin(37°) = 32/sin(B).
   Full step-by-step solution

   Step 1: Identify given values. Side a = 27 m (opposite angle A = 37°), side b = 32 m (opposite angle B). Step 2: Use Law of Sines: a/sin(A) = b/sin(B) → 27/sin(37°) = 32/sin(B). Step 3: Calculate sin(37°) ≈ 0.6018. So 27/0.6018 ≈ 44.87. Step 4: Set up: 44.87 = 32/sin(B) → sin(B) = 32/44.87 ≈ 0.7132. Step 5: Since sin(B) = 0.7132 < 1, two angles satisfy: B₁ ≈ arcsin(0.7132) ≈ 45.5° and B₂ ≈ 180° - 45.5° = 134.5°. Step 6: Check validity with A = 37°: - For B₁ = 45.5°, C₁ = 180° - 37° - 45.5° = 97.5° (valid, positive). - For B₂ = 134.5°, C₂ = 180° - 37° - 134.5° = 8.5° (valid, positive). Step 7: Both triangles satisfy triangle angle sum, so two possible triangles exist. The answer is: Two possible triangles.

8. sin(π/12)cos(π/12) = ? Answer: 1/4 Solution: Use the double-angle identity: sin(2θ) = 2sin(θ)cos(θ) Rearrange to get: sin(θ)cos(θ) = sin(2θ)/2 Apply this identity with θ = π/12 sin(π/12)cos(π/12) = sin(2 × π/12)/2 = sin(π/6)/2 sin(π/6) = 1/2 Therefore, sin(π/12)cos(π/12) = (1/2)/2 = 1/4 The answer is 1/4.
   Full step-by-step solution

   Step 1: Use the double-angle identity: sin(2θ) = 2sin(θ)cos(θ) Step 2: Rearrange to get: sin(θ)cos(θ) = sin(2θ)/2 Step 3: Apply this identity with θ = π/12 Step 4: sin(π/12)cos(π/12) = sin(2 × π/12)/2 = sin(π/6)/2 Step 5: sin(π/6) = 1/2 Step 6: Therefore, sin(π/12)cos(π/12) = (1/2)/2 = 1/4 The answer is 1/4.