Geometry Practice Test: Right Triangle Trigonometry (SOH CAH TOA)
- Right Triangle Trigonometry
- 10 questions
- 14 minutes
Use SOH CAH TOA and a calculator in degree mode. Tip: label the sides opposite, adjacent and hypotenuse from the angle you are working with, not from the right angle.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
Right triangle ABC has its right angle at C, with AC = 7, BC = 24 and AB = 25.
What is tan A?
Correct answer: B
From angle A, the opposite leg is BC = 24 and the adjacent leg is AC = 7, so tan A = 24/7. The ratio 7/24 is tan B.
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Question 2
Right triangle PQR has its right angle at R, with PR = 20, QR = 21 and PQ = 29.
What is cos P?
Correct answer: D
Cosine is adjacent over hypotenuse. The leg next to angle P is PR = 20, so cos P = 20/29. The ratio 21/29 is sin P.
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Question 3
Angle θ is acute and sin θ = 3/5. What is tan θ?
Correct answer: C
Draw a right triangle with opposite side 3 and hypotenuse 5; the adjacent side is 25 − 9 = 4. So tan θ = 3/4. The ratio 4/5 is cos θ.
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Question 4
In a right triangle, one acute angle measures 28° and the hypotenuse is 15 cm long.
To the nearest tenth of a centimeter, how long is the leg opposite the 28° angle?
Correct answer: A
Opposite = hypotenuse · sin 28° = 15 · 0.4695 ≈ 7.0 cm. Using cosine gives 13.2 cm, which is the adjacent leg.
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Question 5
A kite string is 120 feet long and makes a 52° angle with the level ground. Assume the string is straight and is held at ground level.
To the nearest tenth of a foot, how high is the kite?
Correct answer: B
The string is the hypotenuse and the height is opposite the 52° angle: 120 · sin 52° ≈ 94.6 ft. The value 73.9 ft uses cosine and gives the horizontal distance instead.
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Question 6
A ramp rises 5 feet over a horizontal distance of 12 feet.
To the nearest tenth of a degree, what angle does the ramp make with the ground?
Correct answer: D
The rise is opposite the angle and the horizontal run is adjacent, so θ = tan⁻¹(5/12) ≈ 22.6°. Using sin⁻¹(5/12) ≈ 24.6° would treat 12 as the length of the ramp.
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Question 7
When the angle of elevation of the sun is 40°, a tree on level ground casts an 18-meter shadow.
To the nearest tenth of a meter, how tall is the tree?
Correct answer: C
The height is opposite the 40° angle and the shadow is adjacent: h = 18 · tan 40° ≈ 15.1 m. Dividing, 18/tan 40° ≈ 21.5 m, puts the shadow on the wrong side of the ratio.
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Question 8
For an acute angle x, cos x = 0.6. What is sin(90° − x)?
Correct answer: A
The sine of an angle equals the cosine of its complement, so sin(90° − x) = cos x = 0.6. The value 0.8 is sin x, not sin(90° − x).
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Question 9
In a right triangle, one acute angle measures 35° and the leg opposite it is 9 inches long.
To the nearest tenth of an inch, how long is the leg adjacent to the 35° angle?
Correct answer: B
tan 35° = 9/a, so a = 9/tan 35° ≈ 12.9 in. Multiplying 9 · tan 35° ≈ 6.3 in. is the common mistake, and 15.7 in. is the hypotenuse.
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Question 10
A designer wants a ramp that makes a 5° angle with the level ground and rises 30 inches.
To the nearest tenth of an inch, what horizontal distance does the ramp need?
Correct answer: C
The rise is opposite the 5° angle and the horizontal distance is adjacent: d = 30/tan 5° ≈ 342.9 in. The value 344.2 in. is 30/sin 5°, which is the length of the sloped surface.
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