Inverse Trigonometric

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Answer Key & Explanations

Inverse Trigonometric · Grade 11 · Worksheet 3

1. 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.

2. Emma is a civil engineer designing a suspension bridge. The main cable of the bridge follows a parabolic curve and is anchored at two towers that are 200 meters apart. At the lowest point of the cable (the vertex), the cable is 20 meters above the deck of the bridge. The cable is attached to each tower at a height of 60 meters above the deck. Emma needs to calculate the angle θ that the cable makes with the horizontal at the point where it attaches to a tower. The relationship is given by tan(θ) = (vertical rise from vertex to tower) / (half the horizontal span). Using an inverse trigonometric function, what is θ in degrees? Round your answer to one decimal place. Answer: 21.8 Solution: Identify the vertical rise from the lowest point to the tower attachment. The cable is at 20 m above the deck at the vertex and 60 m above the deck at the tower, so the rise is 60 - 20 = 40 meters.
   Full step-by-step solution

   Step 1: Identify the vertical rise from the lowest point to the tower attachment. The cable is at 20 m above the deck at the vertex and 60 m above the deck at the tower, so the rise is 60 - 20 = 40 meters. Step 2: The horizontal span from vertex to tower is half the total span between towers: 200 / 2 = 100 meters. Step 3: Use the tangent function: tan(θ) = opposite/adjacent = vertical rise / horizontal distance = 40/100 = 0.4. Step 4: Apply the inverse tangent function: θ = tan⁻¹(0.4). Step 5: Calculate using a calculator: tan⁻¹(0.4) ≈ 21.801409 degrees. Step 6: Round to one decimal place: 21.8 degrees. The angle the cable makes with the horizontal at the tower is 21.8 degrees.

3. Hana is a marine biologist tracking a pod of dolphins. She observes a dolphin diving from the surface straight down to a depth of 24 meters. From the point directly above the dolphin on the surface, she measures the horizontal distance to a coral formation on the seafloor to be 32 meters. If the dolphin swims in a straight line from its current depth to the coral formation, what angle, in degrees, does its path make with the horizontal seafloor? Use an inverse trigonometric function and round your answer to the nearest tenth of a degree. Answer: 36.9 Solution: Identify the sides of the right triangle. The vertical depth of 24 meters is opposite the angle the path makes with the horizontal. The horizontal distance of 32 meters is adjacent to that angle.
   Full step-by-step solution

   Step 1: Identify the sides of the right triangle. The vertical depth of 24 meters is opposite the angle the path makes with the horizontal. The horizontal distance of 32 meters is adjacent to that angle. Step 2: Use the tangent ratio: tan(theta) = opposite/adjacent = 24/32 = 3/4 = 0.75. Step 3: Apply the inverse tangent function: theta = tan^-1(0.75). Step 4: Using a calculator, tan^-1(0.75) = 36.86989765... degrees. Step 5: Round to the nearest tenth: 36.9 degrees. The angle is 36.9 degrees.

4. Mason is a civil engineer designing a new suspension bridge. The main support cable forms a catenary curve, but for the initial calculations, he approximates it as a straight line from the top of a tower to the anchor point on the ground. The cable makes an angle θ with the horizontal. The horizontal distance from the tower base to the anchor point is 80 meters, and the vertical height of the tower is 30 meters. Later, Mason realizes that the actual angle θ must also satisfy the equation 3 sin(θ) = cos(θ). Using inverse trigonometric functions, determine the angle θ in degrees (between 0° and 90°) that satisfies this equation for the cable's slope. Answer: 18.4 Solution: Start with the equation: 3 sin(θ) = cos(θ) Divide both sides by cos(θ) (valid since cos(θ) > 0 for θ in [0°, 90°]): (3 sin(θ))/cos(θ) = 1 Simplify using the identity tan(θ) = sin(θ)/cos(θ): 3 tan(θ) = 1 Solve for tan(θ): tan(θ) = 1/3 Apply the inverse tangent function: θ = tan⁻¹(1/3) Calculate…
   Full step-by-step solution

   Step 1: Start with the equation: 3 sin(θ) = cos(θ) Step 2: Divide both sides by cos(θ) (valid since cos(θ) > 0 for θ in [0°, 90°]): (3 sin(θ))/cos(θ) = 1 Step 3: Simplify using the identity tan(θ) = sin(θ)/cos(θ): 3 tan(θ) = 1 Step 4: Solve for tan(θ): tan(θ) = 1/3 Step 5: Apply the inverse tangent function: θ = tan⁻¹(1/3) Step 6: Calculate using a calculator: tan⁻¹(1/3) ≈ 18.43494882 degrees Step 7: Round to one decimal place: 18.4 degrees The angle θ that satisfies the equation is approximately 18.4 degrees.

5. 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.