Inverse Trigonometric

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Answer Key & Explanations

Inverse Trigonometric · Grade 11 · Worksheet 2

1. Charlotte is designing a suspension bridge for a school project. The main cable of the bridge forms a parabolic curve, and at a certain point, the vertical distance from the road deck to the cable is 9 meters, while the horizontal distance from the lowest point of the cable to that point is 12 meters. To analyze the cable's tension, Charlotte needs to find the angle θ (in degrees) that the cable makes with the horizontal at that point, where θ satisfies the equation 2 sin(θ) = 3 cos(θ). Solve for θ, where 0° < θ < 90°. Express your answer to the nearest tenth of a degree. Answer: 56.3 Solution: Start with the equation 2 sin(θ) = 3 cos(θ). Divide both sides by cos(θ) (since cos(θ) > 0 for 0° < θ < 90°): 2 sin(θ)/cos(θ) = 3, so 2 tan(θ) = 3. Solve for tan(θ): tan(θ) = 3/2 = 1.5.
   Full step-by-step solution

   Step 1: Start with the equation 2 sin(θ) = 3 cos(θ). Step 2: Divide both sides by cos(θ) (since cos(θ) > 0 for 0° < θ < 90°): 2 sin(θ)/cos(θ) = 3, so 2 tan(θ) = 3. Step 3: Solve for tan(θ): tan(θ) = 3/2 = 1.5. Step 4: Use the inverse tangent function: θ = tan⁻¹(1.5). Step 5: Calculate using a calculator: tan⁻¹(1.5) ≈ 56.30993247 degrees. Step 6: Round to the nearest tenth: 56.3 degrees. The answer is 56.3 degrees.

2. Aroha is an aerospace engineer designing a satellite dish. The dish's cross-section is a parabola. To ensure optimal signal reception, the receiver must be placed at the focus. The depth of the dish is 12 meters, and the diameter is 40 meters. The angle θ that the line from the focus to the rim makes with the axis of symmetry satisfies the equation tan(θ) = (diameter) / (4 × depth). Using inverse trigonometric functions, what is the angle θ in degrees? Round your answer to the nearest tenth of a degree. Answer: 39.8 Solution: Identify the given values: diameter = 40 meters, depth = 12 meters.
   Full step-by-step solution

   Step 1: Identify the given values: diameter = 40 meters, depth = 12 meters. Step 2: Write the formula: tan(θ) = diameter / (4 × depth) Step 3: Substitute the values: tan(θ) = 40 / (4 × 12) = 40 / 48 Step 4: Simplify the fraction: 40/48 = 5/6 ≈ 0.833333... Step 5: Apply the inverse tangent function: θ = arctan(5/6) Step 6: Calculate using a calculator: arctan(5/6) ≈ 39.80557... degrees Step 7: Round to the nearest tenth of a degree: θ ≈ 39.8° The angle θ is approximately 39.8 degrees.

3. An engineer is designing a ramp for wheelchair access to a building. The ramp must rise 2.5 meters vertically over a horizontal distance of 10 meters. What angle should the ramp make with the horizontal ground to meet accessibility standards? Express your answer in degrees to one decimal place. Answer: 14.0° Solution: Vertical rise = 2.5 m Horizontal distance = 10 m We need the angle θ the ramp makes with the horizontal. Identify the right triangle.
   Full step-by-step solution

   We are given: Vertical rise = 2.5 m Horizontal distance = 10 m We need the angle θ the ramp makes with the horizontal. Step 1: Identify the right triangle. The vertical rise is the side opposite angle θ, and the horizontal distance is the side adjacent to angle θ. Step 2: Choose the correct trigonometric ratio. The tangent of θ is opposite over adjacent: tan(θ) = opposite / adjacent = 2.5 / 10 Step 3: Calculate the ratio. 2.5 / 10 = 0.25 So tan(θ) = 0.25 Step 4: Find θ using the inverse tangent function. θ = arctan(0.25) Step 5: Compute arctan(0.25) in degrees. Using a calculator: arctan(0.25) ≈ 14.03624347 degrees Step 6: Round to one decimal place. 14.036... rounds to 14.0 degrees. Step 7: Conclusion. The ramp should make an angle of 14.0° with the horizontal ground. ANSWER: 14.0°

4. A drone is flying at a constant altitude of 120 meters above the ground. The drone operator stands 50 meters from the point directly below the drone. The operator needs to calculate the angle of elevation from her position to the drone. What angle should she calculate? Answer: 67.38° Solution: The drone is at a constant altitude of 120 m above the ground. The operator is 50 m horizontally from the point on the ground directly below the drone. We want the angle of elevation from the operator to the drone.
   Full step-by-step solution

   Let's solve this step by step. --- **Step 1: Understand the problem** The drone is at a constant altitude of 120 m above the ground. The operator is 50 m horizontally from the point on the ground directly below the drone. We want the angle of elevation from the operator to the drone. This forms a right triangle: - Opposite side (vertical) = 120 m (drone altitude) - Adjacent side (horizontal) = 50 m (distance from operator to point below drone) - Hypotenuse = line of sight from operator to drone --- **Step 2: Choose the right trigonometric ratio** The angle of elevation θ is at the operator’s position, between the horizontal and the line of sight to the drone. tan(θ) = Opposite / Adjacent tan(θ) = 120 / 50 --- **Step 3: Calculate the ratio** 120 / 50 = 12 / 5 = 2.4 So: tan(θ) = 2.4 --- **Step 4: Find the angle** θ = arctan(2.4) Using a calculator (set to degrees): arctan(2.4) ≈ 67.380135° --- **Step 5: Round appropriately** The problem’s given correct answer is 67.38°, so we round to two decimal places. --- **Final Answer:** The angle of elevation is 67.38°.

5. Liam is a structural engineer designing a new pedestrian bridge. The bridge will have a main support cable that forms a specific angle with the horizontal deck. For the design to be structurally sound, the angle θ must satisfy the equation 5 sin(θ) = 3 cos(θ), where θ is measured in degrees and lies between 0° and 90°. What angle θ should Liam use for his bridge design? Express your answer to the nearest tenth of a degree. Answer: 31.0 Solution: Start with the equation 5 sin(θ) = 3 cos(θ). Divide both sides by cos(θ) (since cos(θ) is not zero for angles between 0° and 90°): 5 sin(θ)/cos(θ) = 3. Rewrite using the identity tan(θ) = sin(θ)/cos(θ): 5 tan(θ) = 3.
   Full step-by-step solution

   Step 1: Start with the equation 5 sin(θ) = 3 cos(θ). Step 2: Divide both sides by cos(θ) (since cos(θ) is not zero for angles between 0° and 90°): 5 sin(θ)/cos(θ) = 3. Step 3: Rewrite using the identity tan(θ) = sin(θ)/cos(θ): 5 tan(θ) = 3. Step 4: Divide both sides by 5: tan(θ) = 3/5 = 0.6. Step 5: Apply the inverse tangent function: θ = tan⁻¹(0.6). Step 6: Use a calculator to compute tan⁻¹(0.6) in degrees: θ ≈ 30.96375653°. Step 7: Round to the nearest tenth of a degree: θ ≈ 31.0°. The angle Liam should use for his bridge design is 31.0°.

6. arcsin(1/2) + arccos(1/2) = ? Answer: π/2 Solution: We need to find arcsin(1/2) + arccos(1/2). arcsin(1/2) means the angle whose sine is 1/2. arccos(1/2) means the angle whose cosine is 1/2.
   Full step-by-step solution

   Step 1: Understand the problem We need to find arcsin(1/2) + arccos(1/2). arcsin(1/2) means the angle whose sine is 1/2. arccos(1/2) means the angle whose cosine is 1/2. Both angles are in the principal range: arcsin gives angles between -pi/2 and pi/2, arccos gives angles between 0 and pi. Step 2: Find arcsin(1/2) We know sin(pi/6) = 1/2, and pi/6 is within the range of arcsin. So arcsin(1/2) = pi/6. Step 3: Find arccos(1/2) We know cos(pi/3) = 1/2, and pi/3 is within the range of arccos. So arccos(1/2) = pi/3. Step 4: Add the results arcsin(1/2) + arccos(1/2) = pi/6 + pi/3 = pi/6 + 2pi/6 = 3pi/6 = pi/2. Step 5: Final answer pi/2.