Free ACT Math: Trigonometry Questions and Answers 1 — 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)?

5/13
12/13 (Correct answer)
5/12
13/12

Correct answer: 12/13

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?

-1
1 (Correct answer)
√2/2
undefined

Correct answer: 1

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)?

4/5
3/4
-4/5 (Correct answer)
-3/5

Correct answer: -4/5

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)?

π/2 (Correct answer)
4π
2π
3

Correct answer: π/2

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)?

1/2
√3/2
-1/2
-√3/2 (Correct answer)

Correct answer: -√3/2

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.

Question 6: A ladder leans against a vertical wall, making an angle of 60° with the ground. If the base of the ladder is 5 feet from the wall, what is the length of the ladder?

5√3 feet
10 feet (Correct answer)
10√3/3 feet
5/2 feet

Correct answer: 10 feet

This scenario forms a right triangle where the ladder is the hypotenuse, the distance from the wall is the adjacent side, and the wall is the opposite side. We know the adjacent side (5 feet) and the angle (60°), and we want to find the hypotenuse (L). Using SOHCAHTOA, cos(θ) = Adjacent/Hypotenuse. So, cos(60°) = 5/L. Since cos(60°) = 1/2, we have 1/2 = 5/L, which means L = 10 feet.

Question 7: The expression (tan(x) * csc(x)) / sec(x) is equivalent to which of the following?

sin(x)
cos(x)
1 (Correct answer)
tan²(x)

Correct answer: 1

To simplify this expression, convert all functions to sine and cosine. tan(x) = sin(x)/cos(x), csc(x) = 1/sin(x), and sec(x) = 1/cos(x). The expression becomes [(sin(x)/cos(x)) * (1/sin(x))] / [1/cos(x)]. The sin(x) terms in the numerator cancel, leaving [1/cos(x)] / [1/cos(x)], which equals 1.

Free ACT Math: Trigonometry Questions and Answers 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)?