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Trigonometric Ratios Practice

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Trigonometric Ratios Practice

A surveying team is checking a coastal walking track and a lookout platform. Use right-angled trigonometry to calculate missing lengths and angles. Use degree mode, give units, and round lengths to 1 decimal place and angles to 1 decimal place unless stated otherwise.

Choose a ratio

Use SOH–CAH–TOA to decide which ratio or inverse ratio links the known and unknown sides.

1.At a lookout, angle θ is in a right-angled triangle. Relative to θ, the opposite side is 7 m and the adjacent side is 24 m. Which calculation gives θ?

tan⁻¹(7 ÷ 24)
sin⁻¹(7 ÷ 24)
cos⁻¹(7 ÷ 24)
tan⁻¹(24 ÷ 7)

2.A track rises 18 m over a horizontal distance of 45 m. Write the tangent ratio for its angle of elevation, α, before calculating α to 1 decimal place.

3.Complete each ratio definition relative to angle θ: sin θ = /; cos θ = /; tan θ = /.

Track measurements

Sketch each situation if it helps. Identify the right-angled triangle and select a ratio that connects the known and unknown quantities.

4.A viewing platform is 12 m above level ground. From a point 20 m from its base, find the angle of elevation to the platform, to 1 decimal place.

5.A safety cable is 15 m long and makes an angle of 38° with the horizontal ground. How high above the ground is the point where it is attached?

6.A straight ramp has a horizontal run of 4.8 m and an angle of elevation of 9°. Calculate the ramp length to 1 decimal place.

7.A walker sees the top of a 32 m coastal cliff at an angle of elevation of 27°. Assuming level ground and that the cliff height is measured from the walker's eye level, calculate the horizontal distance to the cliff base to the nearest metre.

8.From a lookout 25 m above sea level, the angle of depression to a boat is 14°. Find the boat's horizontal distance from the point directly below the lookout, to 1 decimal place.

Check and explain

Show enough working for another person to follow your reasoning.

9.A right-angled triangle has hypotenuse 13 cm and one acute angle of 52°. Find the side opposite that angle and the side adjacent to it, to 1 decimal place.

10.A student calculates a height using 20 × tan 31°. In a right triangle with angle 31°, the 20 m side is adjacent and the height is opposite. Is the method correct? Explain briefly.

11.A track's gradient is the ratio rise/run = 3/8. Find the angle of elevation to 1 decimal place, then state what happens to the angle if the run stays 8 units but the rise increases to 4 units.