Trigonometry

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Trigonometry · Grade 11 · Worksheet 3

1. Liam is surveying a tall radio tower. From a point 37 meters away from the base of the tower, he measures the angle of elevation to the top as 63°. What is the height of the radio tower? Answer: D. 72.6 meters Solution: In problems involving angles of elevation, we often use right triangle trigonometry. The angle of elevation is measured from the horizontal up to the line of sight. The tangent function relates these two sides: tan(angle) = opposite/adjacent.
   Full step-by-step solution

   In problems involving angles of elevation, we often use right triangle trigonometry. The angle of elevation is measured from the horizontal up to the line of sight. This creates a right triangle where the horizontal distance is adjacent to the angle, and the vertical height is opposite the angle. The tangent function relates these two sides: tan(angle) = opposite/adjacent. When solving similar problems, identify which sides you know and which trigonometric ratio connects them.

2. In a right triangle, if the hypotenuse is 25 and one angle is 40°, find the length of the side adjacent to that angle. Answer: 19.15 Solution: Identify the trigonometric ratio that relates the adjacent side and hypotenuse: cos(angle) = adjacent/hypotenuse Substitute the known values: cos(40°) = adjacent/25 Solve for the adjacent side: adjacent = 25 × cos(40°) Calculate cos(40°) using a calculator: cos(40°) ≈ 0.7660 Multiply: 25 ×…
   Full step-by-step solution

   Step 1: Identify the trigonometric ratio that relates the adjacent side and hypotenuse: cos(angle) = adjacent/hypotenuse Step 2: Substitute the known values: cos(40°) = adjacent/25 Step 3: Solve for the adjacent side: adjacent = 25 × cos(40°) Step 4: Calculate cos(40°) using a calculator: cos(40°) ≈ 0.7660 Step 5: Multiply: 25 × 0.7660 = 19.15 Step 6: The adjacent side is approximately 19.15 units long.

3. In a right triangle, if tan(θ) = 7/9, find sin(θ) and cos(θ). Answer: sin(θ) = 7/√130, cos(θ) = 9/√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. Step 4: cos(θ) = adjacent/hypotenuse = 9/√130. The answer is sin(θ) = 7/√130 and cos(θ) = 9/√130.

4. Sophia is surveying a transmission tower from a distance of 61 meters. She measures the angle of elevation to the top of the tower as 41°. How tall is the transmission tower? Answer: 53 Solution: Identify the right triangle components. The distance from Sophia to the tower (61 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 Sophia to the tower (61 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(41°) = height / 61 Step 4: Solve for the height: height = 61 * tan(41°) Step 5: Calculate tan(41°) using a calculator: tan(41°) ≈ 0.8693 Step 6: Multiply: 61 * 0.8693 ≈ 53.0273 Step 7: Round to the nearest whole number: 53 The transmission tower is 53 meters tall.

5. In a right triangle, if tan(θ) = 15/20, find sin(θ). Answer: 0.6 Solution: Given tan(θ) = 15/20, this means opposite side = 15 and adjacent side = 20. Use the Pythagorean theorem to find the hypotenuse: hypotenuse^2 = 15^2 + 20^2 = 225 + 400 = 625. Hypotenuse = sqrt(625) = 25.
   Full step-by-step solution

   Step 1: Given tan(θ) = 15/20, this means opposite side = 15 and adjacent side = 20. Step 2: Use the Pythagorean theorem to find the hypotenuse: hypotenuse^2 = 15^2 + 20^2 = 225 + 400 = 625. Step 3: Hypotenuse = sqrt(625) = 25. Step 4: sin(θ) = opposite/hypotenuse = 15/25 = 3/5 = 0.6. The answer is 0.6.

6. Aroha is building a wheelchair ramp that must rise 1.5 meters to reach her front door. If the ramp makes a 25° angle with the ground, how long will the ramp need to be? Round your answer to the nearest tenth of a meter. Answer: C. 3.6 m Solution: When solving right triangle problems, identify which sides are given and which one you need to find relative to the given angle.
   Full step-by-step solution

   When solving right triangle problems, identify which sides are given and which one you need to find relative to the given angle. The sine function relates the opposite side to the hypotenuse, the cosine function relates the adjacent side to the hypotenuse, and the tangent function relates the opposite side to the adjacent side. Choose the appropriate trigonometric ratio based on the information you have and what you're trying to find.

7. In a right triangle, if the opposite side is 28 and the adjacent side is 45, find the hypotenuse. Answer: 53 Solution: The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b). Here, the opposite side (a) is 28 and the adjacent side (b) is 45.
   Full step-by-step solution

   Step 1: The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b). Step 2: Here, the opposite side (a) is 28 and the adjacent side (b) is 45. So, c² = a² + b². Step 3: Calculate a²: 28² = 784. Step 4: Calculate b²: 45² = 2025. Step 5: Add the squares: 784 + 2025 = 2809. Step 6: Find the square root to get c: c = √2809 = 53. The hypotenuse is 53.

8. In a right triangle, if the hypotenuse is 25 and one angle is 40°, find the adjacent side. Answer: 19.15 Solution: In a right triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse: cos(angle) = adjacent/hypotenuse. We are given the hypotenuse = 25 and the angle = 40°.
   Full step-by-step solution

   Step 1: In a right triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse: cos(angle) = adjacent/hypotenuse. Step 2: We are given the hypotenuse = 25 and the angle = 40°. Step 3: Set up the equation: cos(40°) = adjacent / 25. Step 4: Solve for the adjacent side: adjacent = 25 * cos(40°). Step 5: Using a calculator, cos(40°) ≈ 0.7660. Step 6: Calculate: adjacent = 25 * 0.7660 = 19.15. The adjacent side is approximately 19.15.

9. In a right triangle, if the side opposite angle A is 7 and the side adjacent to angle A is 11, find tan(A). Answer: 7/11 Solution: In a right triangle, the tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Here, the side opposite angle A is 7, and the side adjacent to angle A is 11.
   Full step-by-step solution

   Step 1: In a right triangle, the tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Step 2: Here, the side opposite angle A is 7, and the side adjacent to angle A is 11. Step 3: Therefore, tan(A) = opposite / adjacent = 7 / 11. The answer is 7/11.