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Law of Sines/Cosines

Grade 11 · Trigonometry · Worksheet 1

  1. Mere is analyzing a triangular sail for a boat. The sail has sides of 44 m and 52 m, and the angle between these sides is 112°. Find the length of the third side of the sail to the nearest meter. Answer: ______________
  2. Hana is a conservation ranger monitoring a rare bird population in a triangular nature reserve. From her observation tower at point A, she spots a nesting site at point B, 14 kilometers away. She also spots a feeding ground at point C, 22 kilometers away. The angle between the lines from the tower to the nesting site and to the feeding ground is 48 degrees. To calculate the flight path distance between the nesting site and the feeding ground for her report, Hana needs to find the straight-line distance from point B to point C. What is this distance, to the nearest kilometer? Answer: ______________
  3. In triangle PQR, side p = 12, side q = 18, and angle R = 110°. Find side r using the Law of Cosines: r = ? Answer: ______________
  4. An engineer is designing a triangular support structure for a bridge. She knows two sides of the triangle are 85 meters and 120 meters long, and the angle opposite the 120-meter side is 42°. To ensure structural integrity, she needs to calculate the measure of the angle opposite the 85-meter side. What is this angle measure to the nearest degree? Answer: ______________
  5. A surveyor needs to determine the distance across a wide river. She stands at point A on one bank and measures a 65° angle to a large tree at point C on the opposite bank. She then walks 120 meters along the riverbank to point B and measures a 40° angle to the same tree. Using the Law of Sines, calculate the distance AC from her original position to the tree across the river. Answer: ______________
  6. sin(45°) / 7 = sin(30°) / a, a = ? Answer: ______________
  7. In triangle ABC, angle A = 35°, angle B = 65°, and side b = 15. Find side a using the Law of Sines: a = ? Answer: ______________
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Answer Key & Explanations

Law of Sines/Cosines · Grade 11 · Worksheet 1

  1. Mere is analyzing a triangular sail for a boat. The sail has sides of 44 m and 52 m, and the angle between these sides is 112°. Find the length of the third side of the sail to the nearest meter. Answer: 80 Solution: Write the Law of Cosines formula: c² = a² + b² - 2ab cos(C) Identify the known values: a = 44 m, b = 52 m, C = 112° Substitute into the formula: c² = 44² + 52² - 2(44)(52) cos(112°) Calculate squares: 44² = 1936, 52² = 2704 Sum of squares: 1936 + 2704 = 4640 Calculate 2ab: 2 × 44 × 52 = 4576…
    Full step-by-step solution

    Step 1: Write the Law of Cosines formula: c² = a² + b² - 2ab cos(C) Step 2: Identify the known values: a = 44 m, b = 52 m, C = 112° Step 3: Substitute into the formula: c² = 44² + 52² - 2(44)(52) cos(112°) Step 4: Calculate squares: 44² = 1936, 52² = 2704 Step 5: Sum of squares: 1936 + 2704 = 4640 Step 6: Calculate 2ab: 2 × 44 × 52 = 4576 Step 7: Find cos(112°). Since 112° = 180° - 68°, cos(112°) = -cos(68°). cos(68°) ≈ 0.374606593, so cos(112°) ≈ -0.374606593 Step 8: Multiply: 4576 × (-0.374606593) ≈ -1714.6 Step 9: Complete the calculation: c² = 4640 - (-1714.6) = 4640 + 1714.6 = 6354.6 Step 10: Take square root: c ≈ √6354.6 ≈ 79.72 Step 11: Round to the nearest meter: 80 m The length of the third side is 80 meters.

  2. Hana is a conservation ranger monitoring a rare bird population in a triangular nature reserve. From her observation tower at point A, she spots a nesting site at point B, 14 kilometers away. She also spots a feeding ground at point C, 22 kilometers away. The angle between the lines from the tower to the nesting site and to the feeding ground is 48 degrees. To calculate the flight path distance between the nesting site and the feeding ground for her report, Hana needs to find the straight-line distance from point B to point C. What is this distance, to the nearest kilometer? Answer: 16 Solution: Identify the known values. Side a = distance from tower to nesting site = 14 km. Side b = distance from tower to feeding ground = 22 km.
    Full step-by-step solution

    Step 1: Identify the known values. Side a = distance from tower to nesting site = 14 km. Side b = distance from tower to feeding ground = 22 km. The included angle C between these two sides is 48 degrees. The unknown side c is the distance from B to 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 = 14^2 + 22^2 - 2(14)(22) cos(48 degrees). Step 4: Calculate the squares: 14^2 = 196, 22^2 = 484. So c^2 = 196 + 484 - 2(14)(22) cos(48 degrees). Step 5: Add the squares: 196 + 484 = 680. So c^2 = 680 - 2(14)(22) cos(48 degrees). Step 6: Compute the product: 2 * 14 * 22 = 616. So c^2 = 680 - 616 cos(48 degrees). Step 7: Find cos(48 degrees). Using a calculator, cos(48 degrees) ≈ 0.6691. Step 8: Multiply: 616 * 0.6691 ≈ 412.1656. Step 9: Subtract: c^2 = 680 - 412.1656 = 267.8344. Step 10: Take the square root: c = sqrt(267.8344) ≈ 16.366. Step 11: Round to the nearest kilometer: c ≈ 16 kilometers. The distance between the nesting site and the feeding ground is approximately 16 kilometers.

  3. In triangle PQR, side p = 12, side q = 18, and angle R = 110°. Find side r using the Law of Cosines: r = ? Answer: 24.8 Solution: Identify the known values. In triangle PQR, side p is opposite angle P, side q is opposite angle Q, and side r is opposite angle R. We are given p = 12, q = 18, and angle R = 110°.
    Full step-by-step solution

    Step 1: Identify the known values. In triangle PQR, side p is opposite angle P, side q is opposite angle Q, and side r is opposite angle R. We are given p = 12, q = 18, and angle R = 110°. The side opposite angle R is r, so we use the Law of Cosines: r² = p² + q² - 2pq cos(R). Step 2: Substitute the known values: r² = 12² + 18² - 2(12)(18) cos(110°). Step 3: Calculate the squares: 12² = 144, 18² = 324. So r² = 144 + 324 - 2(12)(18) cos(110°). Step 4: Multiply: 2(12)(18) = 432. So r² = 468 - 432 cos(110°). Step 5: Find cos(110°). Since 110° is in the second quadrant, cos(110°) = -cos(70°) ≈ -0.3420. Step 6: Substitute: r² = 468 - 432(-0.3420) = 468 + 147.744 = 615.744. Step 7: Take the square root: r = sqrt(615.744) ≈ 24.8. The answer is 24.8.

  4. An engineer is designing a triangular support structure for a bridge. She knows two sides of the triangle are 85 meters and 120 meters long, and the angle opposite the 120-meter side is 42°. To ensure structural integrity, she needs to calculate the measure of the angle opposite the 85-meter side. What is this angle measure to the nearest degree? Answer: 28 Solution: Identify the known values: side a = 120 m, side b = 85 m, angle A = 42° (opposite side a) Apply the Law of Sines: a/sin(A) = b/sin(B) Substitute the known values: 120/sin(42°) = 85/sin(B) Calculate sin(42°) ≈ 0.6691 Set up the equation: 120/0.6691 = 85/sin(B) Calculate 120/0.6691 ≈ 179.3 So…
    Full step-by-step solution

    Step 1: Identify the known values: side a = 120 m, side b = 85 m, angle A = 42° (opposite side a) Step 2: Apply the Law of Sines: a/sin(A) = b/sin(B) Step 3: Substitute the known values: 120/sin(42°) = 85/sin(B) Step 4: Calculate sin(42°) ≈ 0.6691 Step 5: Set up the equation: 120/0.6691 = 85/sin(B) Step 6: Calculate 120/0.6691 ≈ 179.3 Step 7: So 179.3 = 85/sin(B) Step 8: Rearrange: sin(B) = 85/179.3 ≈ 0.4741 Step 9: Find angle B: B = arcsin(0.4741) ≈ 28.3° Step 10: Round to nearest degree: 28° The angle opposite the 85-meter side is 28°.

  5. A surveyor needs to determine the distance across a wide river. She stands at point A on one bank and measures a 65° angle to a large tree at point C on the opposite bank. She then walks 120 meters along the riverbank to point B and measures a 40° angle to the same tree. Using the Law of Sines, calculate the distance AC from her original position to the tree across the river. Answer: Approximately 82.3 meters Solution: The Law of Sines is particularly useful for solving triangles when you know two angles and one side (AAS or ASA cases).
    Full step-by-step solution

    The Law of Sines is particularly useful for solving triangles when you know two angles and one side (AAS or ASA cases). In surveying applications, this method allows calculation of inaccessible distances by creating a triangle with measurable angles and one known side. The constant ratio between sides and their opposite angles provides the mathematical relationship needed to find unknown lengths.

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

    We are given: sin(45°) / 7 = sin(30°) / a 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) / a Step 3: Simplify the left side. (√2 / 2) / 7 = √2 / (2 × 7) = √2 / 14 So the equation becomes: √2 / 14 = (1/2) / a Step 4: Simplify the right side. (1/2) / a = 1 / (2a) So: √2 / 14 = 1 / (2a) Step 5: Cross-multiply. (√2) × (2a) = 14 × 1 2a √2 = 14 Step 6: Solve for a. Divide both sides by 2√2: a = 14 / (2√2) = 7 / √2 Step 7: Rationalize the denominator. Multiply numerator and denominator by √2: a = (7 √2) / (√2 × √2) = (7 √2) / 2 Step 8: Approximate the numerical value. √2 ≈ 1.41421356 So a ≈ (7 × 1.41421356) / 2 = (9.89949492) / 2 ≈ 4.94974746 Step 9: Round to two decimal places. a ≈ 4.95 Final answer: a = 4.95

  7. In triangle ABC, angle A = 35°, angle B = 65°, and side b = 15. Find side a using the Law of Sines: a = ? Answer: 9.5 Solution: Find angle C using the triangle angle sum property: A + B + C = 180° 35° + 65° + C = 180° 100° + C = 180° C = 80° Apply the Law of Sines: a/sin(A) = b/sin(B) a/sin(35°) = 15/sin(65°) a = (15 × sin(35°)) / sin(65°) sin(35°) ≈ 0.5736 sin(65°) ≈ 0.9063 a = (15 × 0.5736) / 0.9063 a = 8.604 / 0.9063…
    Full step-by-step solution

    Step 1: Find angle C using the triangle angle sum property: A + B + C = 180° 35° + 65° + C = 180° 100° + C = 180° C = 80° Step 2: Apply the Law of Sines: a/sin(A) = b/sin(B) a/sin(35°) = 15/sin(65°) Step 3: Solve for a: a = (15 × sin(35°)) / sin(65°) Step 4: Calculate using trigonometric values: sin(35°) ≈ 0.5736 sin(65°) ≈ 0.9063 a = (15 × 0.5736) / 0.9063 a = 8.604 / 0.9063 a ≈ 9.5 The answer is 9.5.