Triangle Trigonometry

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Triangle Trigonometry · Grade 11 · Worksheet 3

1. Emma is designing a triangular garden plot. She measures two sides of the triangle as 9 m and 13 m. The angle between these two sides is 41 degrees. She wants to know the length of the third side. Using the Law of Cosines, calculate the length of the third side, correct to the nearest metre. Answer: 9 Solution: Identify the given values. Side a = 9 m, side b = 13 m, and the included angle C = 41 degrees. We need side c opposite angle C.
   Full step-by-step solution

   Step 1: Identify the given values. Side a = 9 m, side b = 13 m, and the included angle C = 41 degrees. We need side c opposite angle C. Step 2: Apply the Law of Cosines: c^2 = a^2 + b^2 - 2ab cos(C). Step 3: Substitute the values: c^2 = 9^2 + 13^2 - 2(9)(13) cos(41). Step 4: Calculate the squares: 9^2 = 81, 13^2 = 169. So c^2 = 81 + 169 - 2(9)(13) cos(41). Step 5: Calculate 2(9)(13) = 234. So c^2 = 250 - 234 cos(41). Step 6: Find cos(41) using a calculator: cos(41) is approximately 0.7547. Step 7: Multiply: 234 times 0.7547 = 176.5998 (approximately). Step 8: Subtract: 250 - 176.5998 = 73.4002. Step 9: Take the square root: c = sqrt(73.4002) = 8.567 (approximately). Step 10: Round to the nearest metre: 9 m. The length of the third side is 9 m.

2. Sophia is an architect designing a triangular skylight for a modern building. She knows that one side of the triangle measures 26 feet, another side measures 31 feet, and the angle opposite the 31-foot side is 61 degrees. Before finalizing the design, Sophia must determine whether these measurements will form exactly one triangle, two possible triangles, or no triangle at all. How many distinct triangles can be formed with these given measurements? Answer: one triangle Solution: Identify the given values: side a = 31 ft (opposite angle A = 61°), side b = 26 ft (opposite angle B). Step 2: Use the Law of Sines: a/sin(A) = b/sin(B) → 31/sin(61°) = 26/sin(B). Step 3: Calculate sin(61°) ≈ 0.8746.
   Full step-by-step solution

   Step 1: Identify the given values: side a = 31 ft (opposite angle A = 61°), side b = 26 ft (opposite angle B). Step 2: Use the Law of Sines: a/sin(A) = b/sin(B) → 31/sin(61°) = 26/sin(B). Step 3: Calculate sin(61°) ≈ 0.8746. So 31/0.8746 ≈ 35.45. Step 4: Set up the equation: 35.45 = 26/sin(B) → sin(B) = 26/35.45 ≈ 0.7334. Step 5: Since sin(B) = 0.7334 < 1, two angles satisfy this: B₁ = sin⁻¹(0.7334) ≈ 47.1° and B₂ = 180° - 47.1° = 132.9°. Step 6: Check if both are valid with A = 61°: For B₁ = 47.1°, C₁ = 180° - 61° - 47.1° = 71.9° (valid, positive). For B₂ = 132.9°, C₂ = 180° - 61° - 132.9° = -13.9° (invalid, negative angle). Step 7: Only one triangle is possible because the second option leads to a negative angle. The answer is one triangle.

3. sin(π/3) + cos(π/6) = ? Answer: √3 Solution: Recall the value of sin(π/3) We know π/3 radians is 60 degrees. The sine of 60 degrees is √3/2. So, sin(π/3) = √3/2.
   Full step-by-step solution

   Step 1: Recall the value of sin(π/3) We know π/3 radians is 60 degrees. The sine of 60 degrees is √3/2. So, sin(π/3) = √3/2. Step 2: Recall the value of cos(π/6) We know π/6 radians is 30 degrees. The cosine of 30 degrees is √3/2. So, cos(π/6) = √3/2. Step 3: Add the two values sin(π/3) + cos(π/6) = √3/2 + √3/2. Step 4: Combine the terms Both terms have the same denominator, so we add the numerators: (√3 + √3)/2 = (2√3)/2. Step 5: Simplify (2√3)/2 = √3. Final Answer: √3

4. Aroha is a surveyor mapping out a triangular plot of land. She measures one side of the triangle to be 13 meters, another side to be 18 meters, and the angle between these two sides to be 42 degrees. Aroha needs to calculate the area of this triangular plot to determine the amount of fertilizer needed for the soil. What is the area of the triangular plot, rounded to the nearest square meter? Answer: 78 square meters Solution: Identify the given values: side a = 13 meters, side b = 18 meters, included angle C = 42 degrees. The formula for the area of a triangle given two sides and the included angle is: Area = (1/2) * a * b * sin(C).
   Full step-by-step solution

   Step 1: Identify the given values: side a = 13 meters, side b = 18 meters, included angle C = 42 degrees. Step 2: The formula for the area of a triangle given two sides and the included angle is: Area = (1/2) * a * b * sin(C). Step 3: Substitute the values: Area = (1/2) * 13 * 18 * sin(42°). Step 4: Calculate 13 * 18 = 234. Step 5: So, Area = (1/2) * 234 * sin(42°) = 117 * sin(42°). Step 6: Use a calculator to find sin(42°) ≈ 0.6691. Step 7: Multiply: 117 * 0.6691 ≈ 78.2847. Step 8: Round to the nearest square meter: 78 square meters. The area of the triangular plot is approximately 78 square meters.

5. Noah is designing a triangular sail for a small boat. He knows that two sides of the sail measure 11 meters and 16 meters, and the angle opposite the 11-meter side is 26°. Before cutting the fabric, Noah must determine whether these measurements will produce exactly one triangle, two possible triangles, or no triangle at all. How many distinct triangles can be formed from these given measurements? Answer: Two triangles Solution: Identify the given values. Side a = 11 m (opposite angle A = 26°), side b = 16 m (opposite angle B). We need to find angle B using the Law of Sines.
   Full step-by-step solution

   Step 1: Identify the given values. Side a = 11 m (opposite angle A = 26°), side b = 16 m (opposite angle B). We need to find angle B using the Law of Sines. Step 2: Write the Law of Sines: a/sin(A) = b/sin(B) Step 3: Substitute the known values: 11/sin(26°) = 16/sin(B) Step 4: Calculate sin(26°): sin(26°) ≈ 0.4384 Step 5: So 11/0.4384 ≈ 25.09. This is the common ratio. Step 6: Set up equation: 25.09 = 16/sin(B) → sin(B) = 16/25.09 ≈ 0.6377 Step 7: Since sin(B) = 0.6377 < 1, there are two possible angles B that satisfy this: B₁ = arcsin(0.6377) ≈ 39.6° B₂ = 180° - 39.6° = 140.4° Step 8: Check if both angles produce valid triangles with A = 26°: For B₁ = 39.6°: C₁ = 180° - 26° - 39.6° = 114.4° (valid, positive angle) For B₂ = 140.4°: C₂ = 180° - 26° - 140.4° = 13.6° (valid, positive angle) Step 9: Both triangles satisfy the triangle angle sum property, so two distinct triangles can be formed. The answer is two triangles.

6. In triangle ABC, side a = 15, side b = 20, and angle C = 60°. Find side c. Answer: 5√13 Solution: Identify the given values: a = 15, b = 20, angle C = 60°. Substitute the values: c² = 15² + 20² - 2(15)(20) cos(60°). Calculate each term: 15² = 225, 20² = 400, 2(15)(20) = 600, cos(60°) = 1/2.
   Full step-by-step solution

   Step 1: Identify the given values: a = 15, b = 20, angle C = 60°. Step 2: Apply the Law of Cosines: c² = a² + b² - 2ab cos(C). Step 3: Substitute the values: c² = 15² + 20² - 2(15)(20) cos(60°). Step 4: Calculate each term: 15² = 225, 20² = 400, 2(15)(20) = 600, cos(60°) = 1/2. Step 5: Continue calculation: c² = 225 + 400 - 600(1/2) = 625 - 300 = 325. Step 6: Take the square root: c = √325 = √(25 × 13) = 5√13. The answer is 5√13.

7. A triangle has sides of length 8 and 11 with an included angle of 120°. Find the length of the third side. Answer: √273 Solution: Identify the known values: side a = 8, side b = 11, angle C = 120° Apply the Law of Cosines: c² = a² + b² - 2ab cos(C) Substitute the values: c² = 8² + 11² - 2(8)(11) cos(120°) Calculate squares: c² = 64 + 121 - 176 cos(120°) Evaluate cos(120°) = -1/2 Continue calculation: c² = 185 - 176(-1/2) =…
   Full step-by-step solution

   Step 1: Identify the known values: side a = 8, side b = 11, angle C = 120° Step 2: Apply the Law of Cosines: c² = a² + b² - 2ab cos(C) Step 3: Substitute the values: c² = 8² + 11² - 2(8)(11) cos(120°) Step 4: Calculate squares: c² = 64 + 121 - 176 cos(120°) Step 5: Evaluate cos(120°) = -1/2 Step 6: Continue calculation: c² = 185 - 176(-1/2) = 185 + 88 = 273 Step 7: Take square root: c = √273 Step 8: The third side is √273