SAT · Math · Geometry and Trigonometry

Right triangles and trigonometry

Right triangles and trigonometry is part of Geometry and Trigonometry in the SAT 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 missing side with the Pythagorean theorem, sine, cosine, tangent from side lengths, complementary angle identity and special right triangles.

Common traps: treating a leg as the hypotenuse; adding instead of subtracting squares; not taking the square root.

Sample questions

Question 1
A right triangle has legs of lengths 7 and 24. What is the length of the hypotenuse?
Enter your own answer.
Question 2
In a right triangle, the two acute angles are A and B. If sin A = 0.42, what is the value of cos B?
- A.
  0.42
- B.
  0.58
- C.
  0.91
- D.
  2.38
Answer: A. 0.42
A and B are complementary, so cos B = sin(90° − B) = sin A = 0.42 (A). B subtracts 0.42 from 1, which is not a trig identity; C is cos A, √(1 − 0.42²) ≈ 0.91, the cosine of the wrong angle; D is the reciprocal 1/0.42.
Question 3
At the same moment on a sunny afternoon, a student 1.8 metres tall casts a shadow 2.4 metres long, and a flagpole casts a shadow 16 metres long. What is the height, in metres, of the flagpole?
- A.
  12
- B.
  15.4
- C.
  21.3
- D.
  28.8
Answer: A. 12
The sun's rays make similar right triangles, so h/16 = 1.8/2.4 = 0.75 and h = 12 (A). B subtracts the 0.6 m difference instead of using a ratio; C inverts the ratio, 16 × 2.4/1.8 ≈ 21.3; D multiplies 16 by 1.8 and never divides by 2.4.
Question 4
For an acute angle θ in a right triangle, sin θ = 2/3. What is the value of tan θ?
- A.
  √5/3
- B.
  2√5/5
- C.
  √5/2
- D.
  3/2
Answer: B. 2√5/5
Take opposite 2 and hypotenuse 3; the adjacent side is √(9 − 4) = √5, so tan θ = 2/√5 = 2√5/5 (B). A is cos θ; C is adjacent over opposite, the reciprocal of tan θ; D is the hypotenuse over the opposite side.
Question 5
If x = 5π/6, what is the value of (sin x)(cos x)?
- A.
  -√3/2
- B.
  -√3/4
- C.
  -1/4
- D.
  √3/4
Answer: B. -√3/4
At 5π/6 (150°), sin x = 1/2 and cos x = -√3/2, so the product is -√3/4. A is sin 2x, which forgets the factor ½ in sin x cos x = ½ sin 2x; C uses -1/2 for the cosine, a value that belongs to 2π/3; D drops the negative sign from the second quadrant cosine.

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