Triangle Trigonometry

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Triangle Trigonometry · Grade 11 · Worksheet 1

1. 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.

2. In triangle Mere, side a = 16, side b = 22, and angle C = 48°. Find side c. Answer: sqrt(740 - 704*cos(48°)) Solution: Use the Law of Cosines: c² = a² + b² - 2ab·cos(C). Substitute a = 16, b = 22, C = 48°: c² = 16² + 22² - 2(16)(22)·cos(48°). Calculate squares: 16² = 256, 22² = 484.
   Full step-by-step solution

   Step 1: Use the Law of Cosines: c² = a² + b² - 2ab·cos(C). Step 2: Substitute a = 16, b = 22, C = 48°: c² = 16² + 22² - 2(16)(22)·cos(48°). Step 3: Calculate squares: 16² = 256, 22² = 484. Step 4: Sum: 256 + 484 = 740. Step 5: Compute product: 2(16)(22) = 704. Step 6: So c² = 740 - 704·cos(48°). Step 7: Take square root: c = sqrt(740 - 704·cos(48°)). The answer is sqrt(740 - 704·cos(48°)).

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. Mere is a conservation ranger monitoring a triangular wildlife sanctuary. She measures two sides of the sanctuary as 24 kilometers and 37 kilometers, and the angle opposite the 24-kilometer side is 32°. She needs to determine how many possible triangular configurations exist for the sanctuary boundaries based on these measurements. How many distinct triangles can be formed? Answer: 2 Solution: Identify the given values: side a = 24 km (opposite angle A = 32°), side b = 37 km. Use the Law of Sines: a/sin(A) = b/sin(B) → 24/sin(32°) = 37/sin(B). Calculate sin(32°) ≈ 0.5299.
   Full step-by-step solution

   Step 1: Identify the given values: side a = 24 km (opposite angle A = 32°), side b = 37 km. Step 2: Use the Law of Sines: a/sin(A) = b/sin(B) → 24/sin(32°) = 37/sin(B). Step 3: Calculate sin(32°) ≈ 0.5299. Then 24/0.5299 ≈ 45.29. Step 4: Set up equation: 45.29 = 37/sin(B) → sin(B) = 37/45.29 ≈ 0.8171. Step 5: Since sin(B) ≈ 0.8171 < 1, there are two possible angles B: B₁ = arcsin(0.8171) ≈ 54.8°, and B₂ = 180° - 54.8° = 125.2°. Step 6: Check triangle sum: For B₁ = 54.8°, C₁ = 180° - 32° - 54.8° = 93.2° (valid, positive). For B₂ = 125.2°, C₂ = 180° - 32° - 125.2° = 22.8° (valid, positive). Step 7: Both angles produce a valid triangle, so two distinct triangles are possible. The answer is 2.

5. Matiu is a forestry engineer surveying a triangular section of native bush for a conservation project. He measures two sides of the triangle as 58 meters and 73 meters, and the angle opposite the 58-meter side is 41 degrees. Matiu needs to determine how many distinct triangular plots are possible with these measurements before he can finalize the boundary markers. How many distinct triangles can be formed? Answer: two triangles Solution: Identify the given values: side a = 58 m (opposite angle A = 41°), side b = 73 m (opposite angle B). Use the Law of Sines: a/sin(A) = b/sin(B) → 58/sin(41°) = 73/sin(B). Calculate sin(41°) ≈ 0.6561.
   Full step-by-step solution

   Step 1: Identify the given values: side a = 58 m (opposite angle A = 41°), side b = 73 m (opposite angle B). Step 2: Use the Law of Sines: a/sin(A) = b/sin(B) → 58/sin(41°) = 73/sin(B). Step 3: Calculate sin(41°) ≈ 0.6561. So 58/0.6561 ≈ 88.40. Step 4: Set up the equation: 88.40 = 73/sin(B) → sin(B) = 73/88.40 ≈ 0.8258. Step 5: Since sin(B) = 0.8258 < 1, two angles satisfy this: B₁ = sin⁻¹(0.8258) ≈ 55.7° and B₂ = 180° - 55.7° = 124.3°. Step 6: Check if both are valid with A = 41°: - For B₁ = 55.7°, C₁ = 180° - 41° - 55.7° = 83.3° (valid, positive). - For B₂ = 124.3°, C₂ = 180° - 41° - 124.3° = 14.7° (valid, positive). Step 7: Both triangles satisfy the triangle angle sum, so two distinct triangles are possible. The answer is two triangles.

6. Liam is designing a triangular support bracket for a bridge. The bracket has sides of length 8 meters and 11 meters, with an included angle of 60°. He needs to determine the length of the third side to order the correct steel beam. What is the length of the third side of the triangular bracket? Answer: √(185 - 88√3/2) Solution: The Law of Cosines is used to find an unknown side of a triangle when you know two sides and the included angle.
   Full step-by-step solution

   The Law of Cosines is used to find an unknown side of a triangle when you know two sides and the included angle. This formula extends the Pythagorean theorem to non-right triangles and is essential for solving many real-world engineering and design problems involving triangular structures.

7. In triangle ABC, side a = 11, side b = 16, and angle C = 36°. Find side c. Answer: 9.6 Solution: Use the Law of Cosines: c² = a² + b² - 2ab·cos(C) Substitute a = 11, b = 16, C = 36°: c² = 11² + 16² - 2(11)(16)·cos(36°) Compute squares: 11² = 121, 16² = 256 Compute 2ab = 2(11)(16) = 352 cos(36°) ≈ 0.8090 c² = 121 + 256 - 352(0.8090) = 377 - 284.768 = 92.232 Take square root: c = √92.232 ≈…
   Full step-by-step solution

   Step 1: Use the Law of Cosines: c² = a² + b² - 2ab·cos(C) Step 2: Substitute a = 11, b = 16, C = 36°: c² = 11² + 16² - 2(11)(16)·cos(36°) Step 3: Compute squares: 11² = 121, 16² = 256 Step 4: Compute 2ab = 2(11)(16) = 352 Step 5: cos(36°) ≈ 0.8090 Step 6: c² = 121 + 256 - 352(0.8090) = 377 - 284.768 = 92.232 Step 7: Take square root: c = √92.232 ≈ 9.6 The answer is 9.6.