Law of Sines/Cosines

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Law of Sines/Cosines · Grade 11 · Worksheet 2

1. Noah is designing a triangular support frame for a sculpture. He has two steel beams measuring 21 meters and 16 meters, which he plans to join at an angle of 71 degrees. To complete the triangle, he needs to cut a third beam to connect the free ends of the two beams. What is the length of the third beam, to the nearest meter? Answer: 22 Solution: Identify the known values. Let side a = 21 m, side b = 16 m, and the included angle C = 71 degrees. The unknown side is c, opposite angle C.
   Full step-by-step solution

   Step 1: Identify the known values. Let side a = 21 m, side b = 16 m, and the included angle C = 71 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 = 21^2 + 16^2 - 2(21)(16) cos(71 degrees). Step 4: Calculate 21^2 = 441, 16^2 = 256. Step 5: So c^2 = 441 + 256 - 2(21)(16) cos(71 degrees). Step 6: Compute 2(21)(16) = 672. Step 7: So c^2 = 697 - 672 cos(71 degrees). Step 8: Find cos(71 degrees) approximately equals 0.3256. Step 9: Multiply: 672 * 0.3256 approximately equals 218.8032. Step 10: Subtract: 697 - 218.8032 = 478.1968. Step 11: Take the square root: c = sqrt(478.1968) approximately equals 21.868. Step 12: Round to the nearest meter: c approximately equals 22 meters. The length of the third beam is 22 meters.

2. Aroha is a landscape architect designing a triangular pond feature in a city park. She knows two sides of the triangular pond are 47 meters and 33 meters long, and the angle between these two sides is 51 degrees. To order the appropriate length of decorative border for the third side, she needs to calculate its length. What is the length of the third side, to the nearest meter? Answer: 37 Solution: Identify the known values. Side a = 47 m, side b = 33 m, and the included angle C = 51 degrees. The unknown side c is opposite angle C.
   Full step-by-step solution

   Step 1: Identify the known values. Side a = 47 m, side b = 33 m, and the included angle C = 51 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 = 47^2 + 33^2 - 2(47)(33) * cos(51 degrees). Step 4: Calculate 47^2 = 2209, 33^2 = 1089. Step 5: So c^2 = 2209 + 1089 - 2(47)(33) * cos(51 degrees). Step 6: 2209 + 1089 = 3298. Step 7: Compute 2(47)(33) = 3102. Step 8: So c^2 = 3298 - 3102 * cos(51 degrees). Step 9: Find cos(51 degrees) ≈ 0.6293. Step 10: Multiply: 3102 * 0.6293 ≈ 1952.0886. Step 11: Subtract: 3298 - 1952.0886 = 1345.9114. Step 12: Take the square root: c = sqrt(1345.9114) ≈ 36.69. Step 13: Round to the nearest meter: c ≈ 37 meters. The length of the third side is 37 meters.

3. 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.

4. In triangle ABC, side a = 15, side b = 20, and angle C = 45°. Find side c using the Law of Cosines: c = ? Answer: 14.2 Solution: Write the Law of Cosines formula: c² = a² + b² - 2ab cos(C) Substitute the given values: a = 15, b = 20, C = 45° c² = 15² + 20² - 2(15)(20) cos(45°) 15² = 225 20² = 400 2(15)(20) = 600 cos(45°) = √2/2 ≈ 0.7071 c² = 225 + 400 - 600(0.7071) c² = 625 - 424.26 c² = 200.74 c = √200.74 ≈ 14.17 Round…
   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 = 15, b = 20, C = 45° c² = 15² + 20² - 2(15)(20) cos(45°) Step 3: Calculate each term: 15² = 225 20² = 400 2(15)(20) = 600 cos(45°) = √2/2 ≈ 0.7071 Step 4: Substitute and compute: c² = 225 + 400 - 600(0.7071) c² = 625 - 424.26 c² = 200.74 Step 5: Take the square root: c = √200.74 ≈ 14.17 Step 6: Round to one decimal place: c ≈ 14.2 The answer is 14.2.

5. 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.

6. Mere is designing a triangular garden bed. Two sides of the bed measure 42 cm and 57 cm, and the angle between these two sides is 110 degrees. She needs to know the length of the third side to purchase edging material. Using the Law of Cosines, calculate the length of the third side to the nearest centimeter. Answer: 82 Solution: Identify the known values: side a = 42 cm, side b = 57 cm, included angle C = 110 degrees. The unknown side is c.
   Full step-by-step solution

   Step 1: Identify the known values: side a = 42 cm, side b = 57 cm, included angle C = 110 degrees. The unknown side is c. Step 2: Write the Law of Cosines formula: c^2 = a^2 + b^2 - 2ab cos(C) Step 3: Substitute the values: c^2 = 42^2 + 57^2 - 2(42)(57) cos(110°) Step 4: Calculate the squares: 42^2 = 1764, 57^2 = 3249 Step 5: Sum the squares: 1764 + 3249 = 5013 Step 6: Calculate 2ab: 2 × 42 × 57 = 2 × 2394 = 4788 Step 7: Find cos(110°): cos(110°) = cos(180° - 70°) = -cos(70°) ≈ -0.342020143 Step 8: Multiply: 4788 × (-0.342020143) = -1637.592 (approximately) Step 9: Substitute into formula: c^2 = 5013 - (-1637.592) = 5013 + 1637.592 = 6650.592 Step 10: Take the square root: c = sqrt(6650.592) ≈ 81.55 Step 11: Round to the nearest centimeter: 82 cm The third side of the garden bed is approximately 82 cm.

7. A triangular-shaped lake has two sides measuring 280 meters and 320 meters that form a 75° angle. A bridge is planned across the lake connecting the ends of these two sides. Using the Law of Cosines, calculate the exact length of the bridge in meters, expressing your answer in simplest radical form. Answer: √(156800 - 89600√3) Solution: Step 1: Apply the Law of Cosines: c² = a² + b² - 2ab cos(C) Step 2: Substitute the given values: c² = 280² + 320² - 2(280)(320) cos(75°) Step 3: Calculate squares: 280² = 78400, 320² = 102400 Step 4: Sum of squares: 78400 + 102400 = 180800 Step 5: Calculate 2ab: 2 × 280 × 320 = 179200 Step 6:…
   Full step-by-step solution

   Step 1: Apply the Law of Cosines: c² = a² + b² - 2ab cos(C) Step 2: Substitute the given values: c² = 280² + 320² - 2(280)(320) cos(75°) Step 3: Calculate squares: 280² = 78400, 320² = 102400 Step 4: Sum of squares: 78400 + 102400 = 180800 Step 5: Calculate 2ab: 2 × 280 × 320 = 179200 Step 6: cos(75°) = cos(45° + 30°) = cos45°cos30° - sin45°sin30° = (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2)/4 Step 7: Multiply: 179200 × (√6 - √2)/4 = 44800(√6 - √2) Step 8: Substitute into formula: c² = 180800 - 44800(√6 - √2) Step 9: Distribute: c² = 180800 - 44800√6 + 44800√2 Step 10: Factor: c² = 156800 + 44800√2 - 44800√6 Step 11: Factor further: c² = 156800 - 44800(√6 - √2) Step 12: Take square root: c = √[156800 - 44800(√6 - √2)] Step 13: The exact length is √(156800 - 89600√3) meters