Triangle Trigonometry

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Triangle Trigonometry · Grade 11 · Worksheet 2

1. sin(π/3)cos(π/6) + cos(π/3)sin(π/6) = ? Answer: 1 Solution: sin(π/3) cos(π/6) + cos(π/3) sin(π/6) sin(π/3) = sin(60°) = √3 / 2 cos(π/6) = cos(30°) = √3 / 2 cos(π/3) = cos(60°) = 1 / 2 sin(π/6) = sin(30°) = 1 / 2 sin(π/3) cos(π/6) = (√3 / 2) × (√3 / 2) cos(π/3) sin(π/6) = (1 / 2) × (1 / 2) (√3 / 2) × (√3 / 2) = (√3 × √3) / (2 × 2) = 3 / 4 (1 / 2) × (1 /…
   Full step-by-step solution

   Let's solve step by step. We are given: sin(π/3) cos(π/6) + cos(π/3) sin(π/6) --- **Step 1: Recall the exact trigonometric values** sin(π/3) = sin(60°) = √3 / 2 cos(π/6) = cos(30°) = √3 / 2 cos(π/3) = cos(60°) = 1 / 2 sin(π/6) = sin(30°) = 1 / 2 --- **Step 2: Substitute the values into the expression** sin(π/3) cos(π/6) = (√3 / 2) × (√3 / 2) cos(π/3) sin(π/6) = (1 / 2) × (1 / 2) --- **Step 3: Compute each product** First product: (√3 / 2) × (√3 / 2) = (√3 × √3) / (2 × 2) = 3 / 4 Second product: (1 / 2) × (1 / 2) = 1 / 4 --- **Step 4: Add the two results** 3/4 + 1/4 = (3 + 1)/4 = 4/4 = 1 --- **Step 5: Final answer** The expression equals 1. --- **Note:** This also matches the trigonometric identity sin(A)cos(B) + cos(A)sin(B) = sin(A + B) Here A = π/3, B = π/6, so A + B = π/3 + π/6 = π/2 sin(π/2) = 1, which confirms the result.

2. A triangle has sides of length 9 and 12 with an included angle of 60°. Find the length of the third side. Answer: √117 Solution: Identify the given values: side a = 9, side b = 12, angle C = 60° (the included angle) Apply the Law of Cosines: c² = a² + b² - 2ab·cos(C) Substitute the values: c² = 9² + 12² - 2(9)(12)·cos(60°) Calculate: c² = 81 + 144 - 216·(1/2) Simplify: c² = 225 - 108 c² = 117 c = √117 The length of the…
   Full step-by-step solution

   Step 1: Identify the given values: side a = 9, side b = 12, angle C = 60° (the included angle) Step 2: Apply the Law of Cosines: c² = a² + b² - 2ab·cos(C) Step 3: Substitute the values: c² = 9² + 12² - 2(9)(12)·cos(60°) Step 4: Calculate: c² = 81 + 144 - 216·(1/2) Step 5: Simplify: c² = 225 - 108 Step 6: c² = 117 Step 7: c = √117 The length of the third side is √117.

3. sin(3π/4)cos(π/4) + cos(3π/4)sin(π/4) = ? Answer: 0 Solution: Recognize that the expression matches the sine addition formula: sin(A+B) = sinAcosB + cosAsinB Identify A = 3π/4 and B = π/4 Apply the formula: sin(3π/4 + π/4) = sin(π) Calculate 3π/4 + π/4 = 4π/4 = π Evaluate sin(π) = 0 The answer is 0.
   Full step-by-step solution

   Step 1: Recognize that the expression matches the sine addition formula: sin(A+B) = sinAcosB + cosAsinB Step 2: Identify A = 3π/4 and B = π/4 Step 3: Apply the formula: sin(3π/4 + π/4) = sin(π) Step 4: Calculate 3π/4 + π/4 = 4π/4 = π Step 5: Evaluate sin(π) = 0 The answer is 0.

4. Olivia is a land surveyor mapping a triangular plot of land. She measures two sides of the triangle as 50 meters and 70 meters, and the angle opposite the 50-meter side is 35 degrees. Olivia needs to determine whether this configuration results in one possible triangle, two possible triangles, or no triangle at all to proceed with her survey. What is the nature of this triangle configuration? Answer: Two possible triangles Solution: Identify the given values: side a = 50 m (opposite angle A = 35°), side b = 70 m (opposite angle B). Use the Law of Sines to find sin(B): a/sin(A) = b/sin(B) → 50/sin(35°) = 70/sin(B).
   Full step-by-step solution

   Step 1: Identify the given values: side a = 50 m (opposite angle A = 35°), side b = 70 m (opposite angle B). Step 2: Use the Law of Sines to find sin(B): a/sin(A) = b/sin(B) → 50/sin(35°) = 70/sin(B). Step 3: Calculate sin(35°) ≈ 0.5736, so 50/0.5736 ≈ 87.17. Step 4: Set up equation: 87.17 = 70/sin(B) → sin(B) = 70/87.17 ≈ 0.8030. Step 5: Since sin(B) = 0.8030 < 1, two angles satisfy this: B₁ ≈ arcsin(0.8030) ≈ 53.4° and B₂ ≈ 180° - 53.4° = 126.6°. Step 6: Check if both angles are valid with angle A = 35°: - For B₁ = 53.4°, C₁ = 180° - 35° - 53.4° = 91.6° (valid, positive angle). - For B₂ = 126.6°, C₂ = 180° - 35° - 126.6° = 18.4° (valid, positive angle). Step 7: Both triangles satisfy the triangle angle sum property, so two possible triangles exist. The answer is two possible triangles.

5. An engineer is designing a triangular support structure for a bridge. She knows two sides measure 15 meters and 22 meters, and the angle opposite the 15-meter side is 28°. To ensure structural integrity, she needs to determine if this configuration produces one triangle, two possible triangles, or no triangle at all. What is the nature of this triangle configuration? Answer: D. Two possible triangles Solution: Identify the given values: side a = 15 m (opposite angle A), side b = 22 m, angle A = 28° Use Law of Sines to find sin(B): a/sin(A) = b/sin(B) → 15/sin(28°) = 22/sin(B) Calculate sin(28°) ≈ 0.4695, so 15/0.4695 ≈ 31.95 Set up equation: 31.95 = 22/sin(B) → sin(B) = 22/31.95 ≈ 0.6886 Since sin(B)…
   Full step-by-step solution

   Step 1: Identify the given values: side a = 15 m (opposite angle A), side b = 22 m, angle A = 28° Step 2: Use Law of Sines to find sin(B): a/sin(A) = b/sin(B) → 15/sin(28°) = 22/sin(B) Step 3: Calculate sin(28°) ≈ 0.4695, so 15/0.4695 ≈ 31.95 Step 4: Set up equation: 31.95 = 22/sin(B) → sin(B) = 22/31.95 ≈ 0.6886 Step 5: Since sin(B) = 0.6886 < 1, two angles satisfy this: B₁ ≈ 43.5° and B₂ ≈ 180° - 43.5° = 136.5° Step 6: Check if both angles are valid with angle A = 28°: - For B₁ = 43.5°, C₁ = 180° - 28° - 43.5° = 108.5° (valid) - For B₂ = 136.5°, C₂ = 180° - 28° - 136.5° = 15.5° (valid) Step 7: Both triangles satisfy the triangle inequality, so two possible triangles exist. The correct answer is Two possible triangles.

6. Liam is a park ranger designing a triangular hiking trail between three scenic viewpoints. He measures two sides of the triangle: one side is 13 kilometers, another side is 17 kilometers, and the angle opposite the 13-kilometer side is 33 degrees. Liam needs to determine whether these measurements will produce one unique triangle, two possible triangles, or no triangle at all before he can finalize the trail layout. How many distinct triangles can be formed with these given measurements? Answer: two triangles Solution: Identify given values. Side a = 13 km (opposite angle A = 33°), side b = 17 km (opposite angle B). Step 2: Use Law of Sines: a/sin(A) = b/sin(B) → 13/sin(33°) = 17/sin(B).
   Full step-by-step solution

   Step 1: Identify given values. Side a = 13 km (opposite angle A = 33°), side b = 17 km (opposite angle B). Step 2: Use Law of Sines: a/sin(A) = b/sin(B) → 13/sin(33°) = 17/sin(B). Step 3: sin(33°) ≈ 0.5446, so 13/0.5446 ≈ 23.87. Step 4: Set up: 23.87 = 17/sin(B) → sin(B) = 17/23.87 ≈ 0.7122. Step 5: Since sin(B) = 0.7122 < 1, two angles are possible: B₁ = arcsin(0.7122) ≈ 45.4° and B₂ = 180° - 45.4° = 134.6°. Step 6: Check both with A = 33°: For B₁ = 45.4°, C₁ = 180° - 33° - 45.4° = 101.6° (valid). For B₂ = 134.6°, C₂ = 180° - 33° - 134.6° = 12.4° (valid). Both yield positive angles, so two distinct triangles are possible. The answer is two triangles.

7. In triangle ABC, side a = 13, side b = 17, and angle C = 45°. Find side c. Answer: sqrt(458 - 221√2) Solution: Use the Law of Cosines: c² = a² + b² - 2ab·cos(C) Substitute the given values: c² = 13² + 17² - 2(13)(17)·cos(45°) Calculate squares: 13² = 169, 17² = 289 Calculate 2ab: 2(13)(17) = 442 cos(45°) = √2/2 Substitute: c² = 169 + 289 - 442·(√2/2) = 458 - 221√2 Take square root: c = √(458 - 221√2) The…
   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 = 13² + 17² - 2(13)(17)·cos(45°) Step 3: Calculate squares: 13² = 169, 17² = 289 Step 4: Calculate 2ab: 2(13)(17) = 442 Step 5: cos(45°) = √2/2 Step 6: Substitute: c² = 169 + 289 - 442·(√2/2) = 458 - 221√2 Step 7: Take square root: c = √(458 - 221√2) The answer is √(458 - 221√2).