Inverse Trigonometric

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Answer Key & Explanations

Inverse Trigonometric · Grade 11 · Worksheet 1

1. 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.

2. sin(2arctan(4/3)) = ? Answer: 24/25 Solution: Let θ = arctan(4/3). This means tan(θ) = 4/3. Construct a right triangle where the opposite side is 4 and the adjacent side is 3.
   Full step-by-step solution

   Step 1: Let θ = arctan(4/3). This means tan(θ) = 4/3. Step 2: Construct a right triangle where the opposite side is 4 and the adjacent side is 3. The hypotenuse is sqrt(4^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5. Step 3: From the triangle, sin(θ) = opposite/hypotenuse = 4/5 and cos(θ) = adjacent/hypotenuse = 3/5. Step 4: Apply the double-angle formula for sine: sin(2θ) = 2 sin(θ) cos(θ). Step 5: Substitute the values: sin(2θ) = 2 * (4/5) * (3/5) = 24/25. The answer is 24/25.

3. A marine biologist is studying a coral reef formation. From her research boat, she measures that the reef drops vertically 18 meters below the water surface. Using sonar, she determines that the horizontal distance from her boat to the point directly above the base of the reef is 24 meters. What angle of depression, in degrees, should she calculate for her research notes? Express your answer to the nearest tenth of a degree. Answer: 36.9 Solution: Identify the right triangle. The vertical drop of 18 meters is the opposite side, and the horizontal distance of 24 meters is the adjacent side relative to the angle of depression.
   Full step-by-step solution

   Step 1: Identify the right triangle. The vertical drop of 18 meters is the opposite side, and the horizontal distance of 24 meters is the adjacent side relative to the angle of depression. Step 2: Use the tangent function: tan(θ) = opposite/adjacent = 18/24 Step 3: Simplify the fraction: 18/24 = 3/4 = 0.75 Step 4: Apply the inverse tangent function: θ = arctan(0.75) Step 5: Calculate using a calculator: arctan(0.75) ≈ 36.86989765 degrees Step 6: Round to the nearest tenth: 36.9 degrees The angle of depression is 36.9 degrees.

4. A telecommunications engineer is designing a satellite dish with a parabolic reflector. The dish has a focal length of 2.4 meters and a diameter of 8 meters. To properly align the receiver at the focal point, she needs to calculate the angle θ between the axis of symmetry and a line from the focus to the edge of the dish. The relationship is given by tan(θ) = (diameter)/(4 × focal length). What angle θ in degrees should she calculate for this dish? Round your answer to the nearest tenth of a degree. Answer: 39.8 Solution: Identify the given values: focal length = 2.4 m, diameter = 8 m Use the formula: tan(θ) = (diameter)/(4 × focal length) Substitute the values: tan(θ) = 8/(4 × 2.4) = 8/9.6 Simplify: tan(θ) = 8/9.6 = 5/6 ≈ 0.8333 Apply the inverse tangent function: θ = arctan(5/6) ≈ arctan(0.8333) Calculate: θ ≈…
   Full step-by-step solution

   Step 1: Identify the given values: focal length = 2.4 m, diameter = 8 m Step 2: Use the formula: tan(θ) = (diameter)/(4 × focal length) Step 3: Substitute the values: tan(θ) = 8/(4 × 2.4) = 8/9.6 Step 4: Simplify: tan(θ) = 8/9.6 = 5/6 ≈ 0.8333 Step 5: Apply the inverse tangent function: θ = arctan(5/6) ≈ arctan(0.8333) Step 6: Calculate: θ ≈ 39.8056 degrees Step 7: Round to nearest tenth: θ ≈ 39.8 degrees The answer is 39.8 degrees.

5. Matiu is a structural engineer analyzing a new bridge design. The main support cable forms an angle θ with the horizontal deck. The relationship between the vertical tension (in kN) and the horizontal tension (in kN) in the cable is given by the equation: 13 sin θ = 5 cos θ. What is the value of θ, in degrees, between 0° and 90° that satisfies this equation? Use inverse trigonometric functions to solve. Answer: 21.0 Solution: Start with the equation: 13 sin θ = 5 cos θ Divide both sides by cos θ (valid since θ is between 0° and 90°, cos θ ≠ 0): 13 sin θ / cos θ = 5 Simplify: 13 tan θ = 5 Divide both sides by 13: tan θ = 5/13 Apply the inverse tangent function: θ = tan⁻¹(5/13) Calculate using a calculator: θ ≈…
   Full step-by-step solution

   Step 1: Start with the equation: 13 sin θ = 5 cos θ Step 2: Divide both sides by cos θ (valid since θ is between 0° and 90°, cos θ ≠ 0): 13 sin θ / cos θ = 5 Step 3: Simplify: 13 tan θ = 5 Step 4: Divide both sides by 13: tan θ = 5/13 Step 5: Apply the inverse tangent function: θ = tan⁻¹(5/13) Step 6: Calculate using a calculator: θ ≈ tan⁻¹(0.384615...) ≈ 21.0° The angle θ is approximately 21.0°.

6. Kaia is an aerospace engineer designing a landing approach path for a new drone. The drone descends from a cruising altitude of 160 meters to a landing pad located 40 meters horizontally from the point directly below its starting position. To program the autopilot, Kaia needs to determine the angle of descent θ (in degrees) measured from the horizontal. However, the flight computer uses the relationship sin(θ) = 0.25 × cos(θ) to maintain a safe glide slope. What angle θ, between 0° and 90°, satisfies this equation? Express your answer in degrees to one decimal place. Answer: 14.0 Solution: Start with the given equation: sin(θ) = 0.25 × cos(θ) Divide both sides by cos(θ) (since cos(θ) is not zero for θ between 0° and 90°): sin(θ)/cos(θ) = 0.25 Recognize that sin(θ)/cos(θ) = tan(θ): tan(θ) = 0.25 Apply the inverse tangent function: θ = tan⁻¹(0.25) Using a calculator: θ ≈ 14.03624…
   Full step-by-step solution

   Step 1: Start with the given equation: sin(θ) = 0.25 × cos(θ) Step 2: Divide both sides by cos(θ) (since cos(θ) is not zero for θ between 0° and 90°): sin(θ)/cos(θ) = 0.25 Step 3: Recognize that sin(θ)/cos(θ) = tan(θ): tan(θ) = 0.25 Step 4: Apply the inverse tangent function: θ = tan⁻¹(0.25) Step 5: Using a calculator: θ ≈ 14.03624 degrees Step 6: Round to one decimal place: θ ≈ 14.0 degrees The answer is 14.0 degrees.