ACT · Math · Preparing for Higher Math: Geometry

Right triangles and trigonometry

Right triangles and trigonometry is part of Preparing for Higher Math: Geometry in the ACT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.

Question types you will see here include find a side with a trig ratio, trig ratio from side lengths, pythagorean theorem and special right triangles.

Common traps: choosing sine when tangent is needed; multiplying when division is needed; calculator set to the wrong angle mode.

Sample questions

Question 1
A coffee kiosk has a triangular awning in the shape of right triangle ABC with the right angle at C, AB = 26 cm, BC = 10 cm and AC = 24 cm. What is tan B?
- A.
  5/13
- B.
  5/12
- C.
  12/13
- D.
  12/5
Answer: D. 12/5
For angle B, the opposite side is AC = 24 and the adjacent side is BC = 10, so tan B = 24/10 = 12/5. Choice A (5/13) gives sin A or cos B, using the hypotenuse. Choice B (5/12) gives tan A, using the other acute angle. Choice C (12/13) gives sin B, using the hypotenuse.
Question 2
A kite at an Ahmedabad festival is flying 24 meters higher than the spool holder's hand. The taut string makes a 45° angle with the level ground. How many meters of string are out between the hand and the kite?
- A.
  24√2
- B.
  24√3
- C.
  48
- D.
  72
Answer: A. 24√2
The string is the hypotenuse of a right isosceles triangle whose legs are 24 m. A is correct: a 45°-45°-90° triangle has hypotenuse leg × √2 = 24√2 ≈ 34 m. B used the 30°-60°-90° ratio, √3, instead of √2. C doubled the height, which is the hypotenuse rule for a 30° angle, not 45°. D tripled the height with no triangle ratio.
Question 3
In right triangle ABC, angle C is the right angle, BC = 3, AC = 4, and AB = 5. What is the value of sin(A)?
- A.
  3/5
- B.
  3/4
- C.
  4/5
- D.
  5/3
Answer: A. 3/5
For angle A, side BC is opposite and AC is adjacent. sin(A) = opposite/hypotenuse = BC/AB = 3/5.
Question 4
Two straight beams of a rooftop solar frame in Muscat, 10 m and 14 m long, meet at a 60° angle. A cable joins their free ends. To the nearest tenth of a meter, how long is the cable?
- A.
  7.3
- B.
  12.5
- C.
  17.2
- D.
  20.9
Answer: B. 12.5
By the law of cosines, c² = 10² + 14² − 2(10)(14)cos 60° = 156, so c ≈ 12.5 m. A uses cos 30° instead of cos 60°. C drops the cosine term, giving √296. D adds 2ab cos C instead of subtracting it, giving √436.
Question 5
In right triangle ABC, the right angle is at C, AC = 15, BC = 8, and AB = 17. What is sin A?
- A.
  8/17
- B.
  8/15
- C.
  15/17
- D.
  17/8
Answer: A. 8/17
Side BC is opposite angle A and AB is the hypotenuse, so sin A = opposite ÷ hypotenuse = 8/17. Choice B (8/15) is tan A, opposite over adjacent. Choice C (15/17) is cos A, adjacent over hypotenuse. Choice D (17/8) inverts the ratio.

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