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 period of y = 2sin(3x)?

A) 2π/3

B) 6π

C) 3π

D) 2π

Period = 2π/b = 2π/3.

2 / 30

2. What is cos(30°)?

A) 1/2

B) √3/2

C) 1

D) √2/2

cos(30°) = √3/2.

3 / 30

3. What is the inverse sine of 1?

A) 0°

B) 90°

C) 60°

D) 30°

sin⁻¹(1) = 90° because sin(90°) = 1.

4 / 30

4. What is the inverse cosine of (-1)?

A) 0°

B) 270°

C) 180°

D) 90°

cos(180°)=-1. arccos(-1)=180°.

5 / 30

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

A) 1

B) -√3

C) √3

D) -1

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

6 / 30

6. What is sin(210°)?

A) 1/2

B) √3/2

C) −√3/2

D) −1/2

Q3, ref=30°. sin negative. sin(210°)=-1/2.

7 / 30

7. What is sin²(x) + cos²(x)?

A) 2

B) 0

C) 1

D) -1

Explanation: This is the Pythagorean identity — always equals 1

8 / 30

8. What is the reference angle for 330°?

A) 45°

B) 70°

C) 60°

D) 30°

330° is in Q4. Reference angle = 360° - 330° = 30°.

9 / 30

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

A) ½absinC

B) ½abcosC

C) abcosC

D) absinC

Area=½ab sinC.

10 / 30

10. What is the vertical shift of y=sin(x+π/3)-2?

A) π/3

B) −π/3

C) −2

D) 2

Vertical shift=-2. Midline: y=-2.

11 / 30

11. In a right triangle, SOH-CAH-TOA means:

A) sin=hypotenuse/opposite

B) sin=opposite/adjacent

C) sin=adjacent/hypotenuse

D) sin=opposite/hypotenuse

Explanation: SOH = Sine is Opposite over Hypotenuse

12 / 30

12. What is the exact value of cos(π/6)?

A) 1

B) 1/2

C) √2/2

D) √3/2

cos(π/6)=cos(30°)=√3/2.

13 / 30

13. In a right triangle with opposite = 3 and adjacent = 4, what is tan(θ)?

A) 3/5

B) 3/4

C) 4/5

D) 4/3

tan(θ) = opposite/adjacent = 3/4.

14 / 30

14. What is sin(π/3)+cos(π/6)?

A) √3

B) 1

C) √3 and both A and B

D) √3/2+√3/2=√3

sin(60°)=√3/2. cos(30°)=√3/2. Sum=√3.

15 / 30

15. 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.

16 / 30

16. What is the exact value of cos(5π/6)?

A) -√3/2

B) √3/2

C) 1/2

D) -1/2

5π/6=150°. Reference 30°. Q2: cos negative. cos(150°)=-√3/2.

17 / 30

17. If sinθ=cosθ, what are possible values of θ?

A) 30° and 210°

B) 60° and 240°

C) 0° and 180°

D) 45° and 225°

tanθ=1→θ=45° and 225°.

18 / 30

18. What is sin(270°)?

A) 0

B) √2/2

C) 1

D) -1

At 270°, point on unit circle is (0,-1), so sin(270°) = -1.

19 / 30

19. What is sin(5π/6)?

A) −√3/2

B) 1/2

C) √3/2

D) −1/2

5π/6=150°. Q2, ref=30°. sin positive. sin=1/2.

20 / 30

20. What is the exact value of tan(π/4)?

A) √3/2

B) 1

C) √3

D) √2/2

tan(45°)=1.

21 / 30

21. What is the period of the tangent function?

A) 4π

B) π/2

C) π

D) 2π

Period of tan(x) = π.

22 / 30

22. What is the value of sin²(60°)-cos²(30°)?

A) 0

B) 1/2

C) -1/2

D) 1

sin(60°)=√3/2, cos(30°)=√3/2. 3/4-3/4=0.

23 / 30

23. What is sin(60°)?

A) 1

B) √3/2

C) 1/2

D) √2/2

sin(60°) = √3/2.

24 / 30

24. In triangle ABC, a=8, A=45°, B=60°. What is b (using Law of Sines)?

A) 8sin(60°)/sin(45°)

B) sin(60°)/8

C) 8sin(45°)/sin(60°)

D) 8/sin(45°)

a/sinA = b/sinB → b = a×sinB/sinA = 8sin(60°)/sin(45°).

25 / 30

25. A pendulum swings through an angle of π/6 radians. If its length is 9m, what is the arc length?

A) 2π

B) 9π/6=3π/2

C) π

D) 3π/2

s=rθ=9×π/6=3π/2.

26 / 30

26. What is the reference angle for 240°?

A) 40°

B) 60°

C) 50°

D) 30°

240° is in Q3. Reference=240-180=60°.

27 / 30

27. What is tan(π)?

A) 0

B) -1

C) 1

D) Undefined

sin(π)=0, cos(π)=-1. tan=0/(-1)=0.

28 / 30

28. What transformation does y = sin(x) + 3 represent?

A) Vertical shift up 3

B) Horizontal shift right 3

C) Amplitude change to 3

D) Vertical shift down 3

Adding 3 to sin(x) shifts the graph up 3 units.

29 / 30

29. What is cos(0°)?

A) -1

B) 1

C) undefined

D) 0

Explanation: cos(0°) = 1 is a standard trigonometric value

30 / 30

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

A) 90°

B) 30°

C) 45°

D) 60°

tan(60°)=√3.

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