Inverse Trigonometric

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Answer Key & Explanations

Inverse Trigonometric · Grade 11 · Worksheet 1

1. sin⁻¹(1/2) = ? Answer: π/6 Solution: We are asked to find sin⁻¹(1/2).
   Full step-by-step solution

   We are asked to find sin⁻¹(1/2). Step 1: Understand the meaning of sin⁻¹(x). The expression sin⁻¹(1/2) means: "Find the angle θ such that sin(θ) = 1/2, and θ is in the range [-π/2, π/2] for the principal branch of inverse sine." Step 2: Recall common sine values from the unit circle. We know that: sin(π/6) = 1/2 sin(5π/6) = 1/2 Step 3: Check which of these angles lies in the correct range for sin⁻¹. The range for sin⁻¹ is [-π/2, π/2] (or [-90°, 90°] in degrees). - π/6 is about 0.523, which is in [-π/2, π/2]. - 5π/6 is about 2.618, which is not in [-π/2, π/2]. So only π/6 is valid for the principal value. Step 4: Conclude the answer. Since sin(π/6) = 1/2 and π/6 is within the principal range, we have: sin⁻¹(1/2) = π/6. Final answer: π/6

2. Liam is an aerospace engineer designing a landing approach path for a small aircraft. The plane must descend from an altitude of 3500 feet to a runway that is 7000 feet away horizontally. The descent angle θ satisfies the equation 3 tan(θ) = 1.5. What angle θ, in degrees, should Liam use for the approach path? Use an inverse trigonometric function and express your answer to the nearest tenth of a degree. Answer: 26.6 Solution: Start with the equation 3 tan(θ) = 1.5. Divide both sides by 3 to isolate tan(θ): tan(θ) = 1.5 / 3 = 0.5. Use a calculator set to degree mode: tan⁻¹(0.5) ≈ 26.56505118 degrees.
   Full step-by-step solution

   Step 1: Start with the equation 3 tan(θ) = 1.5. Step 2: Divide both sides by 3 to isolate tan(θ): tan(θ) = 1.5 / 3 = 0.5. Step 3: Apply the inverse tangent function to both sides: θ = tan⁻¹(0.5). Step 4: Use a calculator set to degree mode: tan⁻¹(0.5) ≈ 26.56505118 degrees. Step 5: Round to the nearest tenth: 26.6 degrees. The angle θ for the descent path is 26.6 degrees.

3. A 20-foot ladder leans against a building, making an angle θ with the ground. If the base of the ladder is 8 feet from the building, find the measure of angle θ in degrees. Round your answer to the nearest whole number. Answer: 66 Solution: We have a right triangle formed by the ladder, the ground, and the building. Identify the sides relative to angle θ. The ladder is the hypotenuse = 20 ft.
   Full step-by-step solution

   We have a right triangle formed by the ladder, the ground, and the building. Step 1: Identify the sides relative to angle θ. The ladder is the hypotenuse = 20 ft. The base from the building is the adjacent side to θ = 8 ft. Step 2: Choose the correct trigonometric ratio. Since we know the adjacent side and the hypotenuse, we use cosine: cos θ = adjacent / hypotenuse = 8 / 20. Step 3: Simplify the fraction. 8/20 = 2/5 = 0.4. So cos θ = 0.4. Step 4: Find θ using inverse cosine. θ = arccos(0.4). Step 5: Calculate arccos(0.4) in degrees. Using a calculator: arccos(0.4) ≈ 66.42182 degrees. Step 6: Round to the nearest whole number. 66.42182 rounds to 66. Final answer: 66 degrees.

4. sin⁻¹(7/25) + sin⁻¹(24/25) = ? Answer: π/2 Solution: Let α = sin⁻¹(7/25) and β = sin⁻¹(24/25). This means sin(α) = 7/25 and sin(β) = 24/25. For α, construct a right triangle with opposite side 7 and hypotenuse 25.
   Full step-by-step solution

   Step 1: Let α = sin⁻¹(7/25) and β = sin⁻¹(24/25). This means sin(α) = 7/25 and sin(β) = 24/25. Step 2: For α, construct a right triangle with opposite side 7 and hypotenuse 25. The adjacent side is sqrt(25² - 7²) = sqrt(625 - 49) = sqrt(576) = 24. So cos(α) = 24/25. Step 3: For β, construct a right triangle with opposite side 24 and hypotenuse 25. The adjacent side is sqrt(25² - 24²) = sqrt(625 - 576) = sqrt(49) = 7. So cos(β) = 7/25. Step 4: Notice that sin(α) = 7/25 = cos(β) and sin(β) = 24/25 = cos(α). This means α and β are complementary angles: α + β = π/2. Step 5: Therefore, sin⁻¹(7/25) + sin⁻¹(24/25) = π/2. The answer is π/2.

5. Mere is an astronomer observing a meteor streaking across the night sky. She tracks its path and finds that the meteor's altitude angle θ (in degrees) above the horizon satisfies the equation 3 sin(θ) = 2 cos(θ). What is the altitude angle θ, measured in degrees between 0° and 90°, that Mere should record? Express your answer to the nearest tenth of a degree. Answer: 33.7 Solution: Start with the equation 3 sin(θ) = 2 cos(θ). Divide both sides by cos(θ) (which is nonzero in the range 0° to 90°): 3 sin(θ)/cos(θ) = 2. Since sin(θ)/cos(θ) = tan(θ), we get 3 tan(θ) = 2.
   Full step-by-step solution

   Step 1: Start with the equation 3 sin(θ) = 2 cos(θ). Step 2: Divide both sides by cos(θ) (which is nonzero in the range 0° to 90°): 3 sin(θ)/cos(θ) = 2. Step 3: Since sin(θ)/cos(θ) = tan(θ), we get 3 tan(θ) = 2. Step 4: Divide both sides by 3: tan(θ) = 2/3. Step 5: Apply the inverse tangent function: θ = tan⁻¹(2/3). Step 6: Using a calculator, tan⁻¹(2/3) ≈ 33.69006753 degrees. Step 7: Round to the nearest tenth: θ ≈ 33.7 degrees. The answer is 33.7 degrees.

6. Liam is designing a suspension bridge for a science project. The main cable of the bridge forms a curve that can be modeled as a straight line between two towers that are 150 meters apart horizontally. The cable is attached to the top of each tower at a height of 45 meters above the bridge deck. At the lowest point, the cable is 5 meters above the deck. Liam needs to find the angle that the cable makes with the horizontal at the point where it meets the left tower. He recalls that if he considers the right triangle formed by the horizontal distance from the tower to the lowest point (75 meters) and the vertical drop from the tower top to the lowest point (45 - 5 = 40 meters), the angle θ satisfies the equation tan(θ) = 40/75. What is the measure of angle θ in degrees, using the inverse tangent function, rounded to the nearest tenth of a degree? Answer: 28.1 Solution: Identify the right triangle. The horizontal distance from the tower to the lowest point is 75 meters (adjacent side). The vertical drop from the tower top to the lowest point is 45 - 5 = 40 meters (opposite side).
   Full step-by-step solution

   Step 1: Identify the right triangle. The horizontal distance from the tower to the lowest point is 75 meters (adjacent side). The vertical drop from the tower top to the lowest point is 45 - 5 = 40 meters (opposite side). Step 2: Set up the tangent ratio: tan(θ) = opposite/adjacent = 40/75 Step 3: Simplify the fraction: 40/75 = 8/15 ≈ 0.5333... Step 4: Apply the inverse tangent function: θ = tan⁻¹(8/15) Step 5: Use a calculator: θ = tan⁻¹(0.53333...) ≈ 28.0724869 degrees Step 6: Round to the nearest tenth: 28.1 degrees The angle the cable makes with the horizontal is 28.1 degrees.