Inverse Trigonometric

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Answer Key & Explanations

Inverse Trigonometric · Grade 11 · Worksheet 3

1. An engineer is designing a wheelchair ramp that must have a maximum slope of 1:12 according to accessibility guidelines. The ramp needs to rise 2.5 feet to reach a building entrance. What angle, in degrees, should the ramp make with the horizontal ground? Round your answer to the nearest tenth of a degree. Answer: 4.8 Solution: A slope of 1:12 means for every 1 unit of vertical rise, the ramp runs 12 units horizontally. So, rise / run = 1 / 12. Identify the given rise.
   Full step-by-step solution

   Step 1: Understand the slope ratio. A slope of 1:12 means for every 1 unit of vertical rise, the ramp runs 12 units horizontally. So, rise / run = 1 / 12. Step 2: Identify the given rise. The ramp rises 2.5 feet. We don’t yet know the run, but the slope ratio is fixed, so we can find the angle from the slope ratio directly. Step 3: Relate slope to the angle. The slope is equal to tan(θ), where θ is the angle with the horizontal. So, tan(θ) = rise / run = 1 / 12. Step 4: Calculate θ. θ = arctan(1 / 12). First, compute 1 / 12 = 0.083333... Now, arctan(0.083333...) ≈ 4.7636 degrees. Step 5: Round to the nearest tenth. 4.7636 rounds to 4.8 degrees. Step 6: Final check. The slope 1:12 is a standard accessibility slope, and the 2.5 ft rise is not needed for the angle calculation because the slope ratio already determines the angle. Final answer: 4.8 degrees.

2. Aroha is standing at the origin of a coordinate plane. She looks toward a point located at (15, 8). A line segment from the origin to this point forms an angle θ with the positive x-axis. Write an equation for θ using an inverse trigonometric function and solve for the measure of θ in degrees, rounded to one decimal place. Answer: 28.1 Solution: The point (15, 8) gives x = 15 and y = 8. The angle θ is formed between the positive x-axis and the line from the origin to (15, 8).
   Full step-by-step solution

   Step 1: The point (15, 8) gives x = 15 and y = 8. The angle θ is formed between the positive x-axis and the line from the origin to (15, 8). Step 2: The tangent of θ is the ratio of the opposite side (y-coordinate) to the adjacent side (x-coordinate): tan(θ) = 8/15. Step 3: To find θ, use the inverse tangent function: θ = tan⁻¹(8/15). Step 4: Calculate 8/15 ≈ 0.533333. Step 5: Use a calculator to find tan⁻¹(0.533333) ≈ 28.0725 degrees. Step 6: Round to one decimal place: 28.1 degrees. The answer is 28.1.

3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). A line is drawn from the origin to the point (4,3), creating angle θ with the positive x-axis. What is the measure of angle θ in degrees? Use inverse trigonometric functions to find your answer. Answer: 36.87 Solution: We have a right triangle with vertices at (0,0), (4,0), and (4,3). The line from (0,0) to (4,3) is the hypotenuse. Angle θ is at the origin between the positive x-axis and this line.
   Full step-by-step solution

   We have a right triangle with vertices at (0,0), (4,0), and (4,3). The line from (0,0) to (4,3) is the hypotenuse. Angle θ is at the origin between the positive x-axis and this line. Step 1: Identify the sides relative to angle θ. From (0,0): - The horizontal leg goes from (0,0) to (4,0), length = 4. This is the adjacent side to θ. - The vertical leg goes from (4,0) to (4,3), length = 3. But from the origin, the vertical coordinate of (4,3) is 3, so the opposite side length = 3. - The hypotenuse length = sqrt(4^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5. Step 2: Choose a trigonometric function to find θ. We know: tan(θ) = opposite / adjacent = 3/4. So θ = arctan(3/4). Step 3: Calculate arctan(3/4). 3/4 = 0.75. Using a calculator: arctan(0.75) ≈ 36.86989765 degrees. Step 4: Round to two decimal places. 36.86989765 rounds to 36.87 degrees. Final answer: 36.87

4. Matiu is an astronomer observing a star that rises at an angle above the horizon. He knows that the tangent of the star's elevation angle is exactly 0.6. Using inverse trigonometric functions, what is the elevation angle of the star in degrees? (Assume the angle is between 0° and 90°) Answer: 30.96 Solution: We are given that tan(θ) = 0.6, where θ is the elevation angle in degrees.
   Full step-by-step solution

   Step 1: We are given that tan(θ) = 0.6, where θ is the elevation angle in degrees. Step 2: To solve for θ, we apply the inverse tangent function to both sides: θ = tan⁻¹(0.6) Step 3: Using a calculator set to degree mode, we compute: θ ≈ 30.96375653... Step 4: Rounding to two decimal places: θ ≈ 30.96° The elevation angle of the star is approximately 30.96 degrees.

5. Tane is a forest ranger monitoring a wildfire from a lookout tower. He spots a smoke plume at an angle of depression of 17° from the tower. The tower is 45 meters tall. Using inverse trigonometric functions, what is the horizontal distance, in meters, from the base of the tower to the smoke plume? Round your answer to the nearest whole meter. Answer: 147 Solution: The angle of depression from the tower equals the angle of elevation from the ground, which is 17°.
   Full step-by-step solution

   Step 1: The angle of depression from the tower equals the angle of elevation from the ground, which is 17°. The tower height (45 m) is the opposite side, and the horizontal distance (d) is the adjacent side relative to this angle. Step 2: Use the tangent ratio: tan(17°) = opposite/adjacent = 45/d Step 3: Solve for d: d = 45 / tan(17°) Step 4: Calculate tan(17°) ≈ 0.30573068 (using a calculator) Step 5: d = 45 / 0.30573068 ≈ 147.188 meters Step 6: Round to the nearest whole meter: 147 meters The horizontal distance is 147 meters.

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

   Step 1: Let θ = arctan(3/4). This means tan(θ) = 3/4. Step 2: Construct a right triangle where the opposite side is 3 and the adjacent side is 4. The hypotenuse is sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5. Step 3: From the triangle, sin(θ) = opposite/hypotenuse = 3/5 and cos(θ) = adjacent/hypotenuse = 4/5. Step 4: Use the double-angle identity: sin(2θ) = 2 sin(θ) cos(θ). Step 5: Substitute the values: sin(2θ) = 2 * (3/5) * (4/5) = 2 * (12/25) = 24/25. The answer is 24/25.

7. sin(2arctan(5/12)) = ? Answer: 120/169 Solution: Let θ = arctan(5/12). This means tan(θ) = 5/12. Draw a right triangle where the opposite side is 5 and the adjacent side is 12.
   Full step-by-step solution

   Step 1: Let θ = arctan(5/12). This means tan(θ) = 5/12. Step 2: Draw a right triangle where the opposite side is 5 and the adjacent side is 12. Step 3: Calculate the hypotenuse: sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13. Step 4: From the triangle, sin(θ) = 5/13 and cos(θ) = 12/13. Step 5: Apply the double-angle formula: sin(2θ) = 2sin(θ)cos(θ). Step 6: Substitute the values: sin(2θ) = 2 × (5/13) × (12/13) = 2 × 60/169 = 120/169. The answer is 120/169.

8. sin⁻¹(15/17) + sin⁻¹(8/17) = ? Answer: π/2 Solution: Let α = sin⁻¹(15/17) and β = sin⁻¹(8/17). Then sin α = 15/17 and sin β = 8/17. For α, construct a right triangle with opposite side 15 and hypotenuse 17.
   Full step-by-step solution

   Step 1: Let α = sin⁻¹(15/17) and β = sin⁻¹(8/17). Then sin α = 15/17 and sin β = 8/17. Step 2: For α, construct a right triangle with opposite side 15 and hypotenuse 17. The adjacent side is sqrt(17² - 15²) = sqrt(289 - 225) = sqrt(64) = 8. So cos α = 8/17. Step 3: For β, construct a right triangle with opposite side 8 and hypotenuse 17. The adjacent side is sqrt(17² - 8²) = sqrt(289 - 64) = sqrt(225) = 15. So cos β = 15/17. Step 4: Notice that sin α = 15/17 = cos β and sin β = 8/17 = cos α. This means α and β are complementary angles: α + β = π/2. Therefore, sin⁻¹(15/17) + sin⁻¹(8/17) = π/2.