Law of Sines/Cosines

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Law of Sines/Cosines · Grade 11 · Worksheet 2

1. Tane is a pilot flying a triangular search pattern over a forest. He flies from point P to point Q, a distance of 92 kilometers. He then turns and flies from point Q to point R, a distance of 64 kilometers, making an angle of 74 degrees at Q between the two legs of the flight. To complete the pattern, he needs to fly directly from R back to P. How far is the flight from R to P, to the nearest kilometer? Answer: 97 Solution: Identify the known values. Side a = PQ = 92 km, side b = QR = 64 km, and the included angle at Q is C = 74 degrees. The unknown side is c = RP, opposite angle C.
   Full step-by-step solution

   Step 1: Identify the known values. Side a = PQ = 92 km, side b = QR = 64 km, and the included angle at Q is C = 74 degrees. The unknown side is c = RP, opposite angle C. Step 2: Use the Law of Cosines: c^2 = a^2 + b^2 - 2ab cos(C). Step 3: Substitute: c^2 = 92^2 + 64^2 - 2(92)(64) cos(74 degrees). Step 4: Calculate squares: 92^2 = 8464, 64^2 = 4096. So c^2 = 8464 + 4096 - 2(92)(64) cos(74 degrees). Step 5: Compute product: 2 * 92 * 64 = 11776. So c^2 = 12560 - 11776 cos(74 degrees). Step 6: Find cos(74 degrees) ≈ 0.2756. Step 7: Multiply: 11776 * 0.2756 ≈ 3245.7856. Step 8: Subtract: c^2 = 12560 - 3245.7856 = 9314.2144. Step 9: Take square root: c = sqrt(9314.2144) ≈ 96.51. Step 10: Round to nearest kilometer: 97. The distance from R to P is approximately 97 kilometers.

2. Aroha is surveying a triangular field. From a reference point, she measures two sides: one side is 93 meters long, and another side is 115 meters long. The angle between these two sides is 67 degrees. What is the length of the third side of the field, to the nearest meter? Answer: 116 Solution: Write the Law of Cosines: c² = a² + b² - 2ab cos(C), where C is the angle between sides a and b, and c is the side opposite angle C. Identify the given values: a = 93 m, b = 115 m, C = 67°.
   Full step-by-step solution

   Step 1: Write the Law of Cosines: c² = a² + b² - 2ab cos(C), where C is the angle between sides a and b, and c is the side opposite angle C. Step 2: Identify the given values: a = 93 m, b = 115 m, C = 67°. Step 3: Substitute into the formula: c² = 93² + 115² - 2(93)(115) cos(67°). Step 4: Calculate squares: 93² = 8649, 115² = 13225. Step 5: Sum of squares: 8649 + 13225 = 21874. Step 6: Calculate 2ab: 2 × 93 × 115 = 21390. Step 7: Find cos(67°) using a calculator: cos(67°) ≈ 0.390731128. Step 8: Multiply: 21390 × 0.390731128 ≈ 8357.67. Step 9: Subtract: c² = 21874 - 8357.67 = 13516.33. Step 10: Take the square root: c ≈ sqrt(13516.33) ≈ 116.26. Step 11: Round to the nearest meter: 116 m. The length of the third side is 116 meters.

3. sin(30°) × 8 ÷ sin(45°) = ? Answer: 4√2 Solution: Recall the exact trigonometric values. sin(30°) = 1/2 sin(45°) = √2 / 2 Substitute these values into the expression.
   Full step-by-step solution

   Step 1: Recall the exact trigonometric values. We know: sin(30°) = 1/2 sin(45°) = √2 / 2 Step 2: Substitute these values into the expression. The problem is: sin(30°) × 8 ÷ sin(45°) Substituting gives: (1/2) × 8 ÷ (√2 / 2) Step 3: Simplify the multiplication and division. First, multiply (1/2) by 8. (1/2) × 8 = 8/2 = 4 So the expression becomes: 4 ÷ (√2 / 2) Step 4: Divide by a fraction. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of (√2 / 2) is (2 / √2). So, 4 ÷ (√2 / 2) = 4 × (2 / √2) Step 5: Multiply the numbers. 4 × (2 / √2) = (4 × 2) / √2 = 8 / √2 Step 6: Rationalize the denominator. It is standard to remove the square root from the denominator. Multiply both the numerator and denominator by √2: (8 / √2) × (√2 / √2) = (8 × √2) / (√2 × √2) = (8√2) / 2 Step 7: Simplify the fraction. (8√2) / 2 = 4√2 Final Answer: 4√2

4. Two tracking stations, 15 kilometers apart, detect an aircraft. Station A measures the angle of elevation to the aircraft as 38°. Station B measures the angle of elevation as 52°. What is the straight-line distance from Station A to the aircraft, to the nearest kilometer? Answer: 12 Solution: Draw a triangle with vertices at Station A, Station B, and the aircraft. The distance between stations A and B is 15 km. The angle at Station A is 38°, and the angle at Station B is 52°.
   Full step-by-step solution

   Step 1: Draw a triangle with vertices at Station A, Station B, and the aircraft. The distance between stations A and B is 15 km. Step 2: The angle at Station A is 38°, and the angle at Station B is 52°. Find the angle at the aircraft: 180° - 38° - 52° = 90°. Step 3: Now we have a right triangle with the right angle at the aircraft. The side opposite the 38° angle (Station A) is the distance from Station B to the aircraft. The side opposite the 52° angle (Station B) is the distance from Station A to the aircraft, which we want to find. Step 4: Use the Law of Sines: (distance from A to aircraft)/sin(52°) = 15/sin(90°) Step 5: Since sin(90°) = 1, we have: distance from A to aircraft = 15 × sin(52°) Step 6: Calculate sin(52°) ≈ 0.7880 Step 7: Distance = 15 × 0.7880 ≈ 11.82 km Step 8: Round to the nearest kilometer: 12 km The answer is 12.

5. In triangle ABC, side a = 8, side b = 10, and angle C = 60°. Find side c using the Law of Cosines. Answer: √84 Solution: Write the Law of Cosines formula for side c: c² = a² + b² - 2ab cos(C) Substitute the known values: c² = 8² + 10² - 2(8)(10) cos(60°) Calculate the squares: c² = 64 + 100 - 160 cos(60°) Evaluate cos(60°) = 1/2: c² = 64 + 100 - 160(1/2) Simplify: c² = 164 - 80 Calculate: c² = 84 Take the square…
   Full step-by-step solution

   Step 1: Write the Law of Cosines formula for side c: c² = a² + b² - 2ab cos(C) Step 2: Substitute the known values: c² = 8² + 10² - 2(8)(10) cos(60°) Step 3: Calculate the squares: c² = 64 + 100 - 160 cos(60°) Step 4: Evaluate cos(60°) = 1/2: c² = 64 + 100 - 160(1/2) Step 5: Simplify: c² = 164 - 80 Step 6: Calculate: c² = 84 Step 7: Take the square root: c = √84 Step 8: Simplify: c = √(4×21) = 2√21 The answer is √84 or 2√21.

6. Emma is a marine biologist tracking a pod of dolphins. From her observation point at a lighthouse, she spots a dolphin at point A. She measures the distance from the lighthouse to the dolphin as 75 meters, and the distance from the lighthouse to a buoy at point B as 50 meters. The angle between the lines from the lighthouse to the dolphin and to the buoy is 65°. Emma needs to determine the straight-line distance between the dolphin at point A and the buoy at point B to calculate the dolphins' travel path. What is the distance between the dolphin and the buoy, to the nearest meter? Answer: 70 Solution: Identify the known values. Let side a = distance from lighthouse to dolphin = 75 m. Let side b = distance from lighthouse to buoy = 50 m.
   Full step-by-step solution

   Step 1: Identify the known values. Let side a = distance from lighthouse to dolphin = 75 m. Let side b = distance from lighthouse to buoy = 50 m. The included angle C between these two sides is 65°. We need to find side c, the distance between the dolphin and buoy. Step 2: Use the Law of Cosines: c^2 = a^2 + b^2 - 2ab * cos(C). Step 3: Substitute the known values: c^2 = 75^2 + 50^2 - 2(75)(50) * cos(65°). Step 4: Calculate the squares: 75^2 = 5625, 50^2 = 2500. So c^2 = 5625 + 2500 - 2(75)(50) * cos(65°). Step 5: Calculate the product: 2 * 75 * 50 = 7500. So c^2 = 8125 - 7500 * cos(65°). Step 6: Find cos(65°). cos(65°) ≈ 0.4226. Step 7: Multiply: 7500 * 0.4226 ≈ 3169.5. Step 8: Subtract: c^2 = 8125 - 3169.5 = 4955.5. Step 9: Take the square root: c = sqrt(4955.5) ≈ 70.4. Step 10: Round to the nearest meter: 70. The distance between the dolphin and the buoy is approximately 70 meters.

7. sin(30°) × 8 = ? Answer: 4 Solution: Recall the value of sin(30°). The sine of 30 degrees is a standard trigonometric value: sin(30°) = 1/2. Write the given multiplication.
   Full step-by-step solution

   Step 1: Recall the value of sin(30°). The sine of 30 degrees is a standard trigonometric value: sin(30°) = 1/2. Step 2: Write the given multiplication. The problem is: sin(30°) × 8. Substitute the value from Step 1: (1/2) × 8. Step 3: Perform the multiplication. (1/2) × 8 = 8/2. Step 4: Simplify the fraction. 8 divided by 2 equals 4. Final Answer: 4