Trigonometry

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Trigonometry · Grade 11 · Worksheet 1

1. Tane is building a wheelchair ramp that must rise 1.2 meters to reach a doorway. The ramp needs to have a 15° angle of elevation. What length of ramp material should Tane prepare? Answer: A. 4.6 m Solution: In right triangle trigonometry, the sine function relates an angle to the ratio of the opposite side over the hypotenuse. Remember that sin(angle) = opposite/hypotenuse, so you can rearrange this to solve for the hypotenuse when you know the opposite side and the angle.
   Full step-by-step solution

   In right triangle trigonometry, the sine function relates an angle to the ratio of the opposite side over the hypotenuse. When you know the height (opposite side) and need to find the length of the ramp (hypotenuse), you can use the sine ratio. Remember that sin(angle) = opposite/hypotenuse, so you can rearrange this to solve for the hypotenuse when you know the opposite side and the angle.

2. In a right triangle, if tan(θ) = 40/75, find θ to the nearest degree. Answer: 28 Solution: We are given tan(θ) = 40/75 To find θ, use the inverse tangent function: θ = tan⁻¹(40/75) Calculate 40 ÷ 75 = 0.5333 Use a calculator: tan⁻¹(0.5333) ≈ 28.07° Round to the nearest degree: 28° The angle θ is 28°.
   Full step-by-step solution

   Step 1: We are given tan(θ) = 40/75 Step 2: To find θ, use the inverse tangent function: θ = tan⁻¹(40/75) Step 3: Calculate 40 ÷ 75 = 0.5333 Step 4: Use a calculator: tan⁻¹(0.5333) ≈ 28.07° Step 5: Round to the nearest degree: 28° The angle θ is 28°.

3. Isabella is standing 18 meters from the base of a tall building. She looks up at the top of the building at an angle of elevation of 55°. How tall is the building? Round your answer to the nearest meter. Answer: 26 Solution: Identify the right triangle components. The distance from Isabella to the building (18 m) is the adjacent side to the angle of elevation. The building height is the opposite side to the angle.
   Full step-by-step solution

   Step 1: Identify the right triangle components. The distance from Isabella to the building (18 m) is the adjacent side to the angle of elevation. The building height is the opposite side to the angle. Step 2: Use the tangent ratio: tan(angle) = opposite / adjacent Step 3: Substitute the known values: tan(55°) = height / 18 Step 4: Solve for height: height = 18 * tan(55°) Step 5: Calculate tan(55°) using a calculator: tan(55°) ≈ 1.4281 Step 6: Multiply: 18 * 1.4281 ≈ 25.7058 Step 7: Round to the nearest meter: 26 Step 8: The building is approximately 26 meters tall.

4. In a right triangle, if the opposite side is 12 and the adjacent side is 17, find the hypotenuse. Answer: 20.808 Solution: Identify the given sides: opposite = 12, adjacent = 17. Substitute the values: hypotenuse^2 = 12^2 + 17^2. Calculate the squares: hypotenuse^2 = 144 + 289.
   Full step-by-step solution

   Step 1: Identify the given sides: opposite = 12, adjacent = 17. Step 2: Apply the Pythagorean theorem: hypotenuse^2 = opposite^2 + adjacent^2. Step 3: Substitute the values: hypotenuse^2 = 12^2 + 17^2. Step 4: Calculate the squares: hypotenuse^2 = 144 + 289. Step 5: Add the results: hypotenuse^2 = 433. Step 6: Take the square root: hypotenuse = sqrt(433). Step 7: Calculate the square root: hypotenuse ≈ 20.808. The hypotenuse is approximately 20.808.

5. Isabella is setting up a ladder to reach the top of a 27-foot wall. The ladder makes a 62° angle with the ground. How long is the ladder? Round your answer to the nearest foot. Answer: C. 31 feet Solution: In right triangle trigonometry problems, we use the sine, cosine, or tangent ratios depending on which sides and angles we know. The sine ratio relates the opposite side to the hypotenuse (sin = opposite/hypotenuse), the cosine relates the adjacent side to the hypotenuse (cos =…
   Full step-by-step solution

   In right triangle trigonometry problems, we use the sine, cosine, or tangent ratios depending on which sides and angles we know. The sine ratio relates the opposite side to the hypotenuse (sin = opposite/hypotenuse), the cosine relates the adjacent side to the hypotenuse (cos = adjacent/hypotenuse), and the tangent relates the opposite side to the adjacent side (tan = opposite/adjacent). When solving for an unknown side, we set up the appropriate trigonometric equation and solve for the missing value.

6. In a right triangle, if tan(θ) = 7/9, find sin(θ). Answer: 7/√130 Solution: Given tan(θ) = 7/9, this means opposite side = 7 and adjacent side = 9. Use the Pythagorean theorem to find the hypotenuse: hypotenuse = √(7² + 9²) = √(49 + 81) = √130. sin(θ) = opposite/hypotenuse = 7/√130.
   Full step-by-step solution

   Step 1: Given tan(θ) = 7/9, this means opposite side = 7 and adjacent side = 9. Step 2: Use the Pythagorean theorem to find the hypotenuse: hypotenuse = √(7² + 9²) = √(49 + 81) = √130. Step 3: sin(θ) = opposite/hypotenuse = 7/√130. The answer is 7/√130.

7. Aroha stands 42 meters from the base of a tower and measures the angle of elevation to the top as 58°. Find the height of the tower. Answer: 67.2 Solution: Identify the right triangle components. The distance from Aroha to the tower (42 m) is the adjacent side to the angle of elevation. The height of the tower is the opposite side.
   Full step-by-step solution

   Step 1: Identify the right triangle components. The distance from Aroha to the tower (42 m) is the adjacent side to the angle of elevation. The height of the tower is the opposite side. Step 2: Use the tangent ratio: tan(angle) = opposite / adjacent Step 3: Substitute the known values: tan(58°) = height / 42 Step 4: Solve for height: height = 42 × tan(58°) Step 5: Calculate tan(58°) using a calculator: tan(58°) ≈ 1.6003 Step 6: Multiply: 42 × 1.6003 = 67.2126 Step 7: Round to one decimal place: 67.2 meters Therefore, the height of the tower is 67.2 meters.

8. Matiu stands 85 meters from the base of a radio tower and measures the angle of elevation to the top as 48°. Find the height of the tower. Answer: 94.4 Solution: Identify the sides of the right triangle. The distance from Matiu to the tower (85 m) is the adjacent side to the angle of elevation. The height of the tower is the opposite side.
   Full step-by-step solution

   Step 1: Identify the sides of the right triangle. The distance from Matiu to the tower (85 m) is the adjacent side to the angle of elevation. The height of the tower is the opposite side. Step 2: Use the tangent ratio, which is opposite over adjacent: tan(angle) = height / distance. Step 3: Substitute the known values: tan(48°) = height / 85. Step 4: Solve for the height: height = 85 * tan(48°). Step 5: Calculate tan(48°) using a calculator: tan(48°) ≈ 1.1106. Step 6: Multiply: 85 * 1.1106 ≈ 94.401. Step 7: Round to one decimal place: 94.4 meters. The height of the radio tower is approximately 94.4 meters.

9. Emma stands 57 meters from the base of a tower and measures the angle of elevation to the top as 37°. Find the height of the tower. Answer: 43 Solution: Identify the right triangle components. The distance from Emma to the tower (57 m) is the adjacent side to the angle of elevation. The tower height is the opposite side.
   Full step-by-step solution

   Step 1: Identify the right triangle components. The distance from Emma to the tower (57 m) is the adjacent side to the angle of elevation. The tower height is the opposite side. Step 2: Use the tangent ratio: tan(angle) = opposite / adjacent Step 3: Substitute the known values: tan(37°) = height / 57 Step 4: Solve for height: height = 57 * tan(37°) Step 5: Calculate tan(37°) ≈ 0.7536 Step 6: Multiply: 57 * 0.7536 ≈ 42.955 Step 7: Round to the nearest whole number: 43 The height of the tower is approximately 43 meters.