Trigonometry Practice Questions on the ACT

By Lisa Zimmer Hatch, Scott A. Hatch

On the ACT Math Test, the trigonometry questions make up no more than 10 percent of the test, so if you haven’t had trig yet, don’t despair. At least half of the trig questions are very basic, covering trig ratios and basic trig identities.

Here are a couple of sample questions for you to practice.

Sample questions

  1. 1. A right triangle has an angle with a degree measure of 45. Find the tangent of the 45-degree angle.

    (A)

    image0.png

    (B)

    image1.png

    (C)

    image2.png

    (D) 0

    (E) 1

  2. Two similar triangles have bases of 5 and 25. Which of the following expresses the ratio of the areas of the two triangles?

    (F) 1:5

    (G) 1:15

    (H) 1:25

    (J) 1:30

    (K) 1:50

Answers and explanations

  1. 1. The correct answer is Choice (E).

    First draw a 45:45:90 triangle.

    Draw a 45:45:90 triangle.

    Draw a 45:45:90 triangle.

    Remember that the ratio of the sides of this triangle is

    image4.png

    where s stands for the length of the side.

    Then figure out the tan by using the TOA part of SOH CAH TOA.

    image5.png

    If you picked Choice (A), you tried to find the ratio of the hypotenuse to one of the other sides, which isn’t one of the three main trig ratios. Choice (B) is what you’d get if you were asked to find the sine or cosine of the 45-degree angle.

    You know that Choice (C) can’t be correct, because you can’t have a radical in the denominator. If you were asked to find the sine or cosine of the 45-degree angle, you would have to rationalize the final answer:

    image6.png

  2. 2. The correct answer is Choice (H).

    The ratio of the sides is 1:5 or

    image7.png

    The ratio of the areas is the square of the ratio of the sides:

    image8.png

    which is the same as 1:25.

    Bonus: What do you suppose the ratio of the volumes of two similar figures is? Because you find volume in cubic units, the ratio of the volumes of two similar figures is the cube of the ratio of their sides. If Figure A has a base of 5 and similar Figure B has a base of 10, then the ratio of their volumes is

    image9.png