FREE ACT Math: Trigonometry Questions and Answers

Question 1

In the right triangle ΔABC, the right angle is at C. If the length of side AC is 12 and the length of the hypotenuse AB is 13, what is the value of cos(A)?

Accepted Answer

12/13

Answer

SOHCAHTOA is an acronym for the basic trigonometric ratios in a right triangle. Cosine is defined as Adjacent over Hypotenuse (CAH). For angle A, the adjacent side is AC (length 12) and the hypotenuse is AB (length 13). Therefore, cos(A) = 12/13.

Question 2

What is the exact value of tan(5π/4) on the unit circle?

Accepted Answer

1

Answer

The angle 5π/4 is in the third quadrant of the unit circle. The reference angle is π/4. In the third quadrant, both sine and cosine are negative. Since tan(θ) = sin(θ)/cos(θ), the value will be (-√2/2) / (-√2/2), which simplifies to 1.

Question 3

If sin(θ) = 3/5 and θ is in Quadrant II, what is the value of cos(θ)?

Accepted Answer

-4/5

Answer

Use the Pythagorean identity sin²(θ) + cos²(θ) = 1. Substitute sin(θ) = 3/5 to get (3/5)² + cos²(θ) = 1, which is 9/25 + cos²(θ) = 1. Solving for cos²(θ) gives 16/25. Since θ is in Quadrant II, cosine is negative, so cos(θ) = -4/5.

Question 4

What is the period of the function f(x) = 3sin(4x)?

Accepted Answer

π/2

Answer

The period of a sine or cosine function of the form y = Asin(Bx + C) + D is given by the formula 2π/|B|. In this function, B = 4. Therefore, the period is 2π/4, which simplifies to π/2.

Question 5

If sec(θ) = -2 and tan(θ) > 0, what is the value of sin(θ)?

Accepted Answer

-√3/2

Answer

Secant is the reciprocal of cosine, so cos(θ) = -1/2. Cosine is negative in Quadrants II and III. Tangent is positive in Quadrants I and III. The only quadrant that satisfies both conditions is Quadrant III. In Quadrant III, sine is negative. Using sin²(θ) + cos²(θ) = 1, we get sin²(θ) + (-1/2)² = 1, so sin²(θ) = 3/4. Thus, sin(θ) = -√3/2.

ACT Practice Test

ACT Practice Test

FREE ACT Math: Trigonometry Questions and Answers