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Law of Sines/Cosines

Grade 11 · Trigonometry · Worksheet 3

  1. Hana is a surveyor mapping a triangular plot of land for a new community garden. She measures two sides of the triangle: one side is 42 meters long, and another side is 31 meters long. The angle between these two sides is 74 degrees. To complete her map, Hana needs to find the length of the third side of the triangle. What is the length of the third side, to the nearest meter? Answer: ______________
  2. Olivia is designing a triangular hiking trail in a nature reserve. She marks two points, A and B, which are 80 meters apart along a straight path. From point A, she sights a large oak tree at point C, and measures the angle between line AB and line AC as 55 degrees. From point B, she measures the angle between line BA and line BC as 40 degrees. Olivia needs to calculate the distance from point A to the oak tree (side AC) to determine the length of the first leg of the trail. What is the distance AC, to the nearest meter? Answer: ______________
  3. Noah is an architect designing a triangular support truss for a pedestrian bridge. He knows two sides of the triangular truss measure 26 meters and 31 meters, and the angle between these two sides is 56 degrees. To order the correct length of steel for the third side, Noah needs to calculate its exact length. What is the length of the third side of the triangular truss, to the nearest meter? Answer: ______________
  4. sin(45°) / 7 = sin(30°) / x = ? Answer: ______________
  5. sin(45°) × √2 = ? Answer: ______________
  6. Noah is designing a triangular sail for a racing yacht. He has two wooden spars measuring 26 meters and 31 meters, and he knows the angle between them must be 56 degrees for optimal aerodynamics. To cut the third spar correctly, Noah must calculate its exact length. What is the length of the third side of the sail, to the nearest meter? Answer: ______________
  7. sin(30°) × 12 ÷ sin(45°) = ? Answer: ______________
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Answer Key & Explanations

Law of Sines/Cosines · Grade 11 · Worksheet 3

  1. Hana is a surveyor mapping a triangular plot of land for a new community garden. She measures two sides of the triangle: one side is 42 meters long, and another side is 31 meters long. The angle between these two sides is 74 degrees. To complete her map, Hana needs to find the length of the third side of the triangle. What is the length of the third side, to the nearest meter? Answer: 45 Solution: Identify the known values. Let side a = 42 m, side b = 31 m, and the included angle C = 74 degrees. The unknown side is c, opposite angle C.
    Full step-by-step solution

    Step 1: Identify the known values. Let side a = 42 m, side b = 31 m, and the included angle C = 74 degrees. The unknown side is 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 = 42^2 + 31^2 - 2(42)(31) cos(74 degrees). Step 4: Calculate 42^2 = 1764, 31^2 = 961. Step 5: So c^2 = 1764 + 961 - 2(42)(31) cos(74 degrees). Step 6: Compute 2(42)(31) = 2604. Step 7: So c^2 = 2725 - 2604 cos(74 degrees). Step 8: Find cos(74 degrees) using a calculator: cos(74 degrees) is approximately 0.2756. Step 9: Multiply: 2604 * 0.2756 is approximately 717.6624. Step 10: Subtract: 2725 - 717.6624 = 2007.3376. Step 11: Take the square root: c = sqrt(2007.3376) is approximately 44.80. Step 12: Round to the nearest meter: c is approximately 45 meters. The length of the third side is 45 meters.

  2. Olivia is designing a triangular hiking trail in a nature reserve. She marks two points, A and B, which are 80 meters apart along a straight path. From point A, she sights a large oak tree at point C, and measures the angle between line AB and line AC as 55 degrees. From point B, she measures the angle between line BA and line BC as 40 degrees. Olivia needs to calculate the distance from point A to the oak tree (side AC) to determine the length of the first leg of the trail. What is the distance AC, to the nearest meter? Answer: 52 Solution: Identify the known values. In triangle ABC: side AB = 80 m (opposite angle C), angle A = 55 degrees (opposite side a = BC), angle B = 40 degrees (opposite side b = AC). We need side AC, which is opposite angle B.
    Full step-by-step solution

    Step 1: Identify the known values. In triangle ABC: side AB = 80 m (opposite angle C), angle A = 55 degrees (opposite side a = BC), angle B = 40 degrees (opposite side b = AC). We need side AC, which is opposite angle B. Step 2: Find angle C. Sum of angles in a triangle is 180 degrees. C = 180 - 55 - 40 = 85 degrees. Step 3: Apply the Law of Sines: a/sin A = b/sin B = c/sin C. We use the ratio with side AB (c = 80) and its opposite angle C (85 degrees), and side AC (b) with its opposite angle B (40 degrees). So: b / sin(40) = 80 / sin(85). Step 4: Solve for b (AC): b = 80 * sin(40) / sin(85). Step 5: Calculate sin(40) ≈ 0.6428, sin(85) ≈ 0.9962. Step 6: Multiply: 80 * 0.6428 = 51.424. Step 7: Divide: 51.424 / 0.9962 ≈ 51.62. Step 8: Round to the nearest meter: 52. The distance from point A to the oak tree is approximately 52 meters.

  3. Noah is an architect designing a triangular support truss for a pedestrian bridge. He knows two sides of the triangular truss measure 26 meters and 31 meters, and the angle between these two sides is 56 degrees. To order the correct length of steel for the third side, Noah needs to calculate its exact length. What is the length of the third side of the triangular truss, to the nearest meter? Answer: 27 Solution: Identify the known values. Let side a = 26 m, side b = 31 m, and the included angle C = 56 degrees. The unknown side c is opposite angle C.
    Full step-by-step solution

    Step 1: Identify the known values. Let side a = 26 m, side b = 31 m, and the included angle C = 56 degrees. The unknown side c is 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 = 26^2 + 31^2 - 2(26)(31) * cos(56 degrees). Step 4: Calculate the squares: 26^2 = 676, 31^2 = 961. So c^2 = 676 + 961 - 2(26)(31) * cos(56 degrees). Step 5: Add the squares: 676 + 961 = 1637. Step 6: Compute the product: 2 * 26 * 31 = 1612. So c^2 = 1637 - 1612 * cos(56 degrees). Step 7: Find cos(56 degrees). cos(56 degrees) ≈ 0.5592. Step 8: Multiply: 1612 * 0.5592 ≈ 901.4304. Step 9: Subtract: c^2 = 1637 - 901.4304 = 735.5696. Step 10: Take the square root: c = sqrt(735.5696) ≈ 27.12. Step 11: Round to the nearest meter: c ≈ 27 meters. The length of the third side of the triangular truss is approximately 27 meters.

  4. sin(45°) / 7 = sin(30°) / x = ? Answer: 4.95 Solution: sin(45°) / 7 = sin(30°) / x Write down the known sine values. sin(45°) = √2 / 2 sin(30°) = 1/2 Substitute these into the equation. (√2 / 2) / 7 = (1/2) / x Simplify the left side.
    Full step-by-step solution

    We are given the equation: sin(45°) / 7 = sin(30°) / x Step 1: Write down the known sine values. sin(45°) = √2 / 2 sin(30°) = 1/2 Step 2: Substitute these into the equation. (√2 / 2) / 7 = (1/2) / x Step 3: Simplify the left side. (√2 / 2) / 7 = √2 / (2 × 7) = √2 / 14 So we have: √2 / 14 = (1/2) / x Step 4: Simplify the right side. (1/2) / x = 1 / (2x) So now: √2 / 14 = 1 / (2x) Step 5: Cross-multiply. √2 × (2x) = 1 × 14 2x √2 = 14 Step 6: Solve for x. Divide both sides by 2√2: x = 14 / (2√2) x = 7 / √2 Step 7: Rationalize the denominator. Multiply numerator and denominator by √2: x = (7√2) / (√2 × √2) = (7√2) / 2 Step 8: Use the approximate value of √2. √2 ≈ 1.41421356 So x ≈ (7 × 1.41421356) / 2 x ≈ (9.89949492) / 2 x ≈ 4.94974746 Step 9: Round to two decimal places as in the given answer. x ≈ 4.95 Final answer: x = 4.95

  5. sin(45°) × √2 = ? Answer: 1 Solution: Recall the value of sin(45°). sin(45°) = √2 / 2. Write the original expression with this value.
    Full step-by-step solution

    Step 1: Recall the value of sin(45°). sin(45°) = √2 / 2. Step 2: Write the original expression with this value. sin(45°) × √2 = (√2 / 2) × √2. Step 3: Multiply the terms. (√2 / 2) × √2 = (√2 × √2) / 2. Step 4: Simplify √2 × √2. √2 × √2 = 2. Step 5: Substitute back into the expression. (√2 × √2) / 2 = 2 / 2. Step 6: Simplify the fraction. 2 / 2 = 1. Final Answer: 1

  6. Noah is designing a triangular sail for a racing yacht. He has two wooden spars measuring 26 meters and 31 meters, and he knows the angle between them must be 56 degrees for optimal aerodynamics. To cut the third spar correctly, Noah must calculate its exact length. What is the length of the third side of the sail, to the nearest meter? Answer: 27 Solution: Identify known values. Let side a = 26 m, side b = 31 m, and the included angle C = 56 degrees. The unknown side c is opposite angle C.
    Full step-by-step solution

    Step 1: Identify known values. Let side a = 26 m, side b = 31 m, and the included angle C = 56 degrees. The unknown side c is 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 = 26^2 + 31^2 - 2(26)(31) cos(56 degrees). Step 4: Calculate the squares: 26^2 = 676, 31^2 = 961. So c^2 = 676 + 961 - 2(26)(31) cos(56 degrees). Step 5: Add: 676 + 961 = 1637. So c^2 = 1637 - 2(26)(31) cos(56 degrees). Step 6: Compute 2(26)(31) = 1612. So c^2 = 1637 - 1612 cos(56 degrees). Step 7: Find cos(56 degrees). Using a calculator, cos(56 degrees) ≈ 0.5592. Step 8: Multiply: 1612 * 0.5592 ≈ 901.4304. Step 9: Subtract: c^2 = 1637 - 901.4304 = 735.5696. Step 10: Take the square root: c = sqrt(735.5696) ≈ 27.12. Step 11: Round to the nearest meter: c ≈ 27 meters. The length of the third side is 27 meters.

  7. sin(30°) × 12 ÷ sin(45°) = ? Answer: 6√2 Solution: Write the expression: sin(30°) × 12 ÷ sin(45°) Substitute known values: sin(30°) = 1/2, sin(45°) = √2/2 Substitute: (1/2) × 12 ÷ (√2/2) Simplify: (1/2) × 12 = 6 Now we have 6 ÷ (√2/2) = 6 × (2/√2) Simplify: 6 × (2/√2) = 12/√2 Rationalize the denominator: (12/√2) × (√2/√2) = 12√2/2 Simplify:…
    Full step-by-step solution

    Step 1: Write the expression: sin(30°) × 12 ÷ sin(45°) Step 2: Substitute known values: sin(30°) = 1/2, sin(45°) = √2/2 Step 3: Substitute: (1/2) × 12 ÷ (√2/2) Step 4: Simplify: (1/2) × 12 = 6 Step 5: Now we have 6 ÷ (√2/2) = 6 × (2/√2) Step 6: Simplify: 6 × (2/√2) = 12/√2 Step 7: Rationalize the denominator: (12/√2) × (√2/√2) = 12√2/2 Step 8: Simplify: 12√2/2 = 6√2 The answer is 6√2.