Law of Sines/Cosines

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Law of Sines/Cosines · Grade 11 · Worksheet 3

1. Mason is a marine archaeologist mapping a submerged triangular artifact site near a coral reef. Using sonar from his boat at point A, he measures the distance to a large rock formation at point B as 92 meters. He then travels to point C and measures the distance from point C back to the boat at point A as 64 meters. The angle at point A between the lines to points B and C is 54°. Mason needs to know the straight-line distance between the rock formation at point B and point C to complete his site map. What is the distance between points B and C, to the nearest meter? Answer: 75 Solution: Identify the known values. Side a = distance from boat at A to rock formation at B = 92 m. Side b = distance from boat at A to point C = 64 m.
   Full step-by-step solution

   Step 1: Identify the known values. Side a = distance from boat at A to rock formation at B = 92 m. Side b = distance from boat at A to point C = 64 m. The included angle C between these two sides is 54°. We need to find side c, the distance between points B and C. Step 2: Use the Law of Cosines: c^2 = a^2 + b^2 - 2ab * cos(C). Step 3: Substitute the known values: c^2 = 92^2 + 64^2 - 2(92)(64) * cos(54°). Step 4: Calculate the squares: 92^2 = 8464, 64^2 = 4096. So c^2 = 8464 + 4096 - 2(92)(64) * cos(54°). Step 5: Add: 8464 + 4096 = 12560. Step 6: Calculate the product: 2 * 92 * 64 = 11776. So c^2 = 12560 - 11776 * cos(54°). Step 7: Find cos(54°). cos(54°) ≈ 0.5878. Step 8: Multiply: 11776 * 0.5878 ≈ 6922.5728. Step 9: Subtract: c^2 = 12560 - 6922.5728 = 5637.4272. Step 10: Take the square root: c = sqrt(5637.4272) ≈ 75.08. Step 11: Round to the nearest meter: 75. The distance between points B and C is approximately 75 meters.

2. Mason is surveying a triangular piece of land for a construction project. He measures two sides of the triangle: one side is 18 meters long, another side is 24 meters long, and the angle opposite the 18-meter side is 35 degrees. Determine the length of the third side to the nearest meter. Answer: 31 Solution: Use Law of Sines to find angle B opposite side b = 24 m. a/sin A = b/sin B -> 18/sin(35) = 24/sin B -> sin B = 24 * sin(35) / 18.
   Full step-by-step solution

   Step 1: Use Law of Sines to find angle B opposite side b = 24 m. a/sin A = b/sin B -> 18/sin(35) = 24/sin B -> sin B = 24 * sin(35) / 18. sin(35) ≈ 0.5736 -> sin B = 24 * 0.5736 / 18 = 13.7664 / 18 ≈ 0.7648 -> B ≈ arcsin(0.7648) ≈ 49.9 degrees (acute case). Step 2: Find angle C: C = 180 - A - B = 180 - 35 - 49.9 = 95.1 degrees. Step 3: Use Law of Cosines to find side c opposite angle C: c^2 = a^2 + b^2 - 2ab cos C = 18^2 + 24^2 - 2*18*24*cos(95.1). cos(95.1) ≈ -0.0906. c^2 = 324 + 576 - 864*(-0.0906) = 900 + 78.2784 = 978.2784. c ≈ sqrt(978.2784) ≈ 31.28. Rounded to nearest meter: 31 meters.

3. 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

4. Hana is a marine archaeologist exploring a submerged triangular reef system. From her research vessel at point P, she measures the distance to a coral formation at point Q as 62 meters, and the distance to a second coral formation at point R as 47 meters. The angle at point P between the lines to Q and R is 52 degrees. Hana needs to know the straight-line distance between the two coral formations (from Q to R) to plan her diving route. What is the distance between Q and R, to the nearest meter? Answer: 50 Solution: Identify the known values. Side PQ = 62 m, side PR = 47 m, and the included angle at P = 52 degrees. Let side QR = c, which is opposite angle P.
   Full step-by-step solution

   Step 1: Identify the known values. Side PQ = 62 m, side PR = 47 m, and the included angle at P = 52 degrees. Let side QR = c, which is opposite angle P. Step 2: Apply the Law of Cosines: c^2 = a^2 + b^2 - 2ab * cos(C), where a = 62, b = 47, and C = 52 degrees. Step 3: Substitute the values: c^2 = 62^2 + 47^2 - 2(62)(47) * cos(52 degrees). Step 4: Calculate the squares: 62^2 = 3844, 47^2 = 2209. So c^2 = 3844 + 2209 - 2(62)(47) * cos(52 degrees). Step 5: Sum the squares: 3844 + 2209 = 6053. So c^2 = 6053 - 2(62)(47) * cos(52 degrees). Step 6: Calculate the product: 2 * 62 * 47 = 5828. So c^2 = 6053 - 5828 * cos(52 degrees). Step 7: Find cos(52 degrees). cos(52 degrees) ≈ 0.6157. Step 8: Multiply: 5828 * 0.6157 ≈ 3588.3. Step 9: Subtract: c^2 = 6053 - 3588.3 = 2464.7. Step 10: Take the square root: c = sqrt(2464.7) ≈ 49.65. Step 11: Round to the nearest meter: 50. The distance between the two coral formations is approximately 50 meters.

5. Noah is an architect designing a triangular plaza. He knows two sides of the plaza measure 56 meters and 81 meters, and the angle opposite the 81-meter side is 71°. To complete his blueprints, Noah needs to find the measure of the angle opposite the 56-meter side. What is this angle measure, to the nearest degree? Answer: 41 Solution: Label the triangle. Let side a = 56 m, side b = 81 m, and angle B = 71° (opposite side b). We need angle A, opposite side a.
   Full step-by-step solution

   Step 1: Label the triangle. Let side a = 56 m, side b = 81 m, and angle B = 71° (opposite side b). We need angle A, opposite side a. Step 2: Apply the Law of Sines: a/sin A = b/sin B. Step 3: Substitute the known values: 56/sin A = 81/sin 71°. Step 4: Cross-multiply: 56 * sin 71° = 81 * sin A. Step 5: Solve for sin A: sin A = (56 * sin 71°) / 81. Step 6: Find sin 71°. sin 71° ≈ 0.9455. Step 7: Multiply: 56 * 0.9455 = 52.948. Step 8: Divide: 52.948 / 81 ≈ 0.6537. Step 9: Find angle A: A = sin⁻¹(0.6537) ≈ 40.8°. Step 10: Since angle B is 71°, and the sum of angles in a triangle is 180°, angle A must be acute (less than 90°), so we take the acute angle. Rounding to the nearest degree gives 41°. The measure of the angle opposite the 56-meter side is 41°.

6. 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 150 meters along the riverbank to point B and measures a 40° angle to the same tree. Using the Law of Sines, calculate the distance from point B to the tree across the river. Answer: Approximately 217.3 meters Solution: The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
   Full step-by-step solution

   The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. This is particularly useful in surveying and navigation problems where you can measure angles but not directly measure distances. When you know two angles and one side, you can determine all other sides of the triangle.