Master ACT Trigonometry Quiz for Exam Success

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30 random questions |30 minutes | Questions are randomized every attempt. Good luck!

ACT Math - Trigonometry Practice Test

Test your ACT trigonometry skills with 30 random practice test questions covering sin, cos, tan, and identities!

1 / 30

1. What is the amplitude of y = -3sin(x)?

A) -3

B) 3

C) 1

D) 6

Amplitude = |-3| = 3. Amplitude is always positive.

2 / 30

2. What is the inverse tangent of 1?

A) 60°

B) 30°

C) 45°

D) 90°

tan⁻¹(1) = 45° because tan(45°) = 1.

3 / 30

3. In a right triangle with angle 60° and adjacent side 5, what is the hypotenuse?

A) 10

B) 10√3

C) 5√3

D) 5/2

cos(60°)=1/2 = adjacent/hypotenuse = 5/h → h = 10.

4 / 30

4. What is the value of sin²(30°) + cos²(30°)?

A) 1

B) 2

C) √3/2

D) 1/2

By the Pythagorean identity, sin²θ + cos²θ = 1 always.

5 / 30

5. What is the period of y = sin(x/2)?

A) 4π

B) 2π

C) π

D) 8π

Period = 2π/b = 2π/(1/2) = 4π.

6 / 30

6. What is cos(360°)?

A) 0

B) 180

C) −1

D) 1

cos(360°)=cos(0°)=1.

7 / 30

7. From a 100m cliff, angle of depression to a boat is 45°. How far is the boat?

A) 100/√2m

B) 100√2m

C) 50m

D) 100m

tan(45°)=100/d=1. d=100m.

8 / 30

8. What is the Law of Sines used for?

A) Right triangles only

B) Solving non-right triangles with angle-side pairs

C) Solving systems of equations

D) Finding area of triangle

Law of Sines solves triangles when you have angle-side pairs.

9 / 30

9. A right triangle has legs 6 and 8. What is tan of the angle opposite the side of length 6?

A) 3/4

B) 3/5

C) 4/5

D) 4/3

tan = opposite/adjacent = 6/8 = 3/4.

10 / 30

10. What is 90° in radians?

A) π

B) π/4

C) 2π

D) π/2

90° × π/180 = π/2 radians.

11 / 30

11. What is the value of (sinθ+cosθ)² when sinθcosθ=1/4?

A) 3/2

B) 1.5

C) Both A and B

D) 2

sin²θ+2sinθcosθ+cos²θ=1+2(1/4)=3/2.

12 / 30

12. What is sin(180°)?

A) 0

B) 1

C) -1

D) undefined

At 180° on unit circle, point is (-1,0), so sin(180°) = 0.

13 / 30

13. What is the value of sin(90°-θ)?

A) -cosθ

B) cosθ

C) -sinθ

D) sinθ

Co-function: sin(90°-θ)=cosθ.

14 / 30

14. What is cos(7π/6)?

A) −1/2

B) 1/2

C) √3/2

D) −√3/2

7π/6=210°. Q3. cos negative. ref=30°. cos=-√3/2.

15 / 30

15. What is the value of sin(θ)×csc(θ)?

A) 0

B) csc²θ

C) 1

D) sin²θ

sinθ×1/sinθ=1.

16 / 30

16. What is the exact value of tan(135°)?

A) √3

B) 1

C) -√3

D) -1

135° in Q2. Reference=45°. tan negative in Q2. tan(135°)=-1.

17 / 30

17. What is the cotangent of 30°?

A) 1/√3

B) √3

C) √3/3

D) 1

cot(30°) = cos(30°)/sin(30°) = (√3/2)/(1/2) = √3.

18 / 30

18. A wheel of radius 2m rotates at 3 radians/sec. What is the linear speed?

A) 2π m/s

B) 6 m/s

C) 9 m/s

D) 3 m/s

v=ωr=3×2=6 m/s.

19 / 30

19. What is the amplitude of y = -6cos(x)?

A) 1

B) 6

C) 12

D) -6

Amplitude=|-6|=6.

20 / 30

20. What is the value of csc(90°)?

A) 0

B) 1

C) Undefined

D) √2

csc=1/sin. sin(90°)=1. csc(90°)=1.

21 / 30

21. What is the value of 1+tan²(30°)?

A) 2

B) 4

C) 3/4

D) 4/3

tan(30°)=1/√3. tan²(30°)=1/3. 1+1/3=4/3.

22 / 30

22. What is the exact value of sin(2π/3)?

A) 1/2

B) -1/2

C) -√3/2

D) √3/2

2π/3=120°. Q2, reference 60°. sin positive. sin(120°)=√3/2.

23 / 30

23. What is sin(π)?

A) π

B) 0

C) 1

D) -1

sin(π)=sin(180°)=0.

24 / 30

24. What is sin(3π/2)?

A) 1

B) 0

C) √2/2

D) −1

sin(270°)=-1.

25 / 30

25. A 20-foot ramp makes a 15° angle with the ground. How high does it rise?

A) 20 cos(15°)

B) 20 tan(15°)

C) 20/sin(15°)

D) 20 sin(15°)

Height = hypotenuse × sin(angle) = 20sin(15°).

26 / 30

26. What is the formula for the area of a triangle given two sides and included angle?

A) ½absinC

B) ½abcosC

C) absinC

D) abcosC

Area=½ab sinC.

27 / 30

27. What is the value of sec(0°)?

A) 0

B) Undefined

C) 1

D) -1

sec=1/cos. cos(0°)=1. sec(0°)=1.

28 / 30

28. What is the value of 2cos²(45°)?

A) √2

B) 2

C) 1/2

D) 1

cos(45°)=√2/2. cos²(45°)=1/2. 2×1/2=1.

29 / 30

29. What is the amplitude of y=3cos(2x)+1?

A) 2

B) 4

C) 1

D) 3

Amplitude=|3|=3.

30 / 30

30. If tan(θ)=√3 and θ is acute, what is θ?

A) 45°

B) 90°

C) 60°

D) 30°

tan(60°)=√3.

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