Law of Sines/Cosines

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Answer Key & Explanations

Law of Sines/Cosines · Grade 11 · Worksheet 1

1. Matiu is designing a triangular garden bed. He knows two sides of the triangular bed are 24 meters and 18 meters long, and the angle between these two sides is 68 degrees. To purchase enough edging for the third side, he needs to calculate its length. What is the length of the third side, to the nearest meter? Answer: 24 Solution: Identify the known values. Side a = 24 m, side b = 18 m, and the included angle C = 68 degrees. The unknown side is c, opposite angle C.
   Full step-by-step solution

   Step 1: Identify the known values. Side a = 24 m, side b = 18 m, and the included angle C = 68 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 = 24^2 + 18^2 - 2(24)(18) cos(68 degrees). Step 4: Calculate 24^2 = 576, 18^2 = 324. Step 5: So c^2 = 576 + 324 - 2(24)(18) cos(68 degrees). Step 6: Compute 2(24)(18) = 864. Step 7: So c^2 = 900 - 864 cos(68 degrees). Step 8: Find cos(68 degrees) ≈ 0.3746. Step 9: Multiply: 864 * 0.3746 ≈ 323.6544. Step 10: Subtract: 900 - 323.6544 = 576.3456. Step 11: Take the square root: c = sqrt(576.3456) ≈ 24.007. Step 12: Round to the nearest meter: c ≈ 24 meters. The length of the third side is 24 meters.

2. In triangle ABC, side a = 15, side b = 20, and angle C = 60°. Find side c using the Law of Cosines: c = ? Answer: 18.0 Solution: Write the Law of Cosines for side c: c² = a² + b² - 2ab cos(C) Substitute a = 15, b = 20, and C = 60°: c² = 15² + 20² - 2(15)(20) cos(60°) Calculate squares: 15² = 225, 20² = 400, so c² = 225 + 400 - 2(15)(20) cos(60°) Multiply: 2(15)(20) = 600, so c² = 625 - 600 cos(60°) cos(60°) = 0.5, so 600…
   Full step-by-step solution

   Step 1: Write the Law of Cosines for side c: c² = a² + b² - 2ab cos(C) Step 2: Substitute a = 15, b = 20, and C = 60°: c² = 15² + 20² - 2(15)(20) cos(60°) Step 3: Calculate squares: 15² = 225, 20² = 400, so c² = 225 + 400 - 2(15)(20) cos(60°) Step 4: Multiply: 2(15)(20) = 600, so c² = 625 - 600 cos(60°) Step 5: cos(60°) = 0.5, so 600 × 0.5 = 300 Step 6: c² = 625 - 300 = 325 Step 7: Take square root: c = sqrt(325) = sqrt(25 × 13) = 5 × sqrt(13) ≈ 5 × 3.6056 = 18.028 Step 8: Round to one decimal: c ≈ 18.0 The answer is 18.0.

3. In triangle ABC, side a = 20, side b = 25, and angle C = 120°. Find side c using the Law of Cosines. Answer: 39.1 Solution: Write the Law of Cosines formula: c² = a² + b² - 2ab cos C Substitute the given values: a = 20, b = 25, C = 120° c² = 20² + 25² - 2(20)(25) cos(120°) 20² = 400 25² = 625 2(20)(25) = 1000 cos(120°) = -1/2 Substitute: c² = 400 + 625 - 1000 × (-1/2) c² = 1025 - (-500) c² = 1025 + 500 c² = 1525 Take…
   Full step-by-step solution

   Step 1: Write the Law of Cosines formula: c² = a² + b² - 2ab cos C Step 2: Substitute the given values: a = 20, b = 25, C = 120° c² = 20² + 25² - 2(20)(25) cos(120°) Step 3: Calculate each term: 20² = 400 25² = 625 2(20)(25) = 1000 cos(120°) = -1/2 Step 4: Substitute: c² = 400 + 625 - 1000 × (-1/2) c² = 1025 - (-500) c² = 1025 + 500 c² = 1525 Step 5: Take the square root: c = sqrt(1525) Simplify: sqrt(1525) = sqrt(25 × 61) = 5 sqrt(61) Approximate: 5 × 7.810 = 39.05 ≈ 39.1 The answer is 39.1.

4. 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.

5. In triangle ABC, side a = 13, side b = 17, and angle C = 55°. Find side c using the Law of Cosines. Answer: 14.3 Solution: The Law of Cosines states: c² = a² + b² - 2ab cos(C). Here, side c is opposite angle C, and sides a and b are the other two sides. Substitute a = 13, b = 17, and C = 55°: c² = 13² + 17² - 2(13)(17) cos(55°).
   Full step-by-step solution

   Step 1: The Law of Cosines states: c² = a² + b² - 2ab cos(C). Here, side c is opposite angle C, and sides a and b are the other two sides. Step 2: Substitute a = 13, b = 17, and C = 55°: c² = 13² + 17² - 2(13)(17) cos(55°). Step 3: Calculate squares: 13² = 169, 17² = 289. So c² = 169 + 289 - 2(13)(17) cos(55°). Step 4: Multiply: 2(13)(17) = 442. So c² = 458 - 442 cos(55°). Step 5: Find cos(55°) ≈ 0.5736. Then 442 × 0.5736 ≈ 253.5. Step 6: Subtract: c² = 458 - 253.5 = 204.5. Step 7: Take square root: c = sqrt(204.5) ≈ 14.3. The answer is 14.3.

6. In triangle PQR, side p = 14, side q = 19, and angle R = 115°. Find side r using the Law of Cosines: r = ? Answer: 28.0 Solution: Write the Law of Cosines for side r: r² = p² + q² - 2pq cos(R) Substitute p = 14, q = 19, and R = 115°: r² = 14² + 19² - 2(14)(19) cos(115°) Calculate squares: 14² = 196, 19² = 361, so r² = 196 + 361 - 2(14)(19) cos(115°) Multiply: 2(14)(19) = 532, so r² = 557 - 532 cos(115°) cos(115°) ≈…
   Full step-by-step solution

   Step 1: Write the Law of Cosines for side r: r² = p² + q² - 2pq cos(R) Step 2: Substitute p = 14, q = 19, and R = 115°: r² = 14² + 19² - 2(14)(19) cos(115°) Step 3: Calculate squares: 14² = 196, 19² = 361, so r² = 196 + 361 - 2(14)(19) cos(115°) Step 4: Multiply: 2(14)(19) = 532, so r² = 557 - 532 cos(115°) Step 5: cos(115°) ≈ -0.4226, so 532 × (-0.4226) = -224.8232 Step 6: r² = 557 - (-224.8232) = 557 + 224.8232 = 781.8232 Step 7: Take square root: r = sqrt(781.8232) ≈ 27.96 Step 8: Round to one decimal: r ≈ 28.0 The answer is 28.0.

7. Hana is designing a triangular garden. Two sides of the garden measure 44 meters and 68 meters, and the angle between these two sides is 52 degrees. She wants to install a straight fence along the third side. Determine the length of this fence, rounding your answer to the nearest meter. Answer: 54 Solution: Identify the known values. We have a triangle with sides a = 44 m, b = 68 m, and the included angle C = 52 degrees. We need to find side c.
   Full step-by-step solution

   Step 1: Identify the known values. We have a triangle with sides a = 44 m, b = 68 m, and the included angle C = 52 degrees. We need to find side c. Step 2: Apply the Law of Cosines: c^2 = a^2 + b^2 - 2ab cos(C) Step 3: Substitute the values: c^2 = 44^2 + 68^2 - 2(44)(68) cos(52) Step 4: Calculate the squares: 44^2 = 1936, 68^2 = 4624 Step 5: Sum the squares: 1936 + 4624 = 6560 Step 6: Calculate 2ab: 2 * 44 * 68 = 5984 Step 7: Find cos(52 degrees) using a calculator: cos(52) ≈ 0.615661475 Step 8: Multiply: 5984 * 0.615661475 ≈ 3684.13 Step 9: Complete the calculation: c^2 = 6560 - 3684.13 = 2875.87 Step 10: Take the square root: c = sqrt(2875.87) ≈ 53.63 Step 11: Round to the nearest meter: 54 meters. The length of the fence is 54 meters.

8. A surveyor needs to determine the distance across a wide river. From point A on one bank, she measures an angle of 68° to a tree at point C on the opposite bank. She then walks 150 meters along the riverbank to point B and measures an angle of 42° back to the same tree. Using the Law of Sines, calculate the distance across the river from point A to the tree at point C. Answer: Approximately 112.3 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 relationship allows us to find unknown side lengths when we know two angles and one side, or two sides and one angle. In surveying applications, this principle is commonly used to measure inaccessible distances by creating triangles with measurable components.