Practice Problems
Practice problems to help review trigonometric concepts.
Section 1: Evaluate Trigonometric Functions
Evaluate: sin(π/6)
Evaluate: cos(3π/4)
Evaluate: tan(5π/3)
Evaluate: sec(π/3)
Evaluate: csc(7π/6)
Evaluate: cot(π/4)
Section 2: Right-Triangle Trigonometry
A right triangle has opposite = 9 and hypotenuse = 15. Find sin θ.
A right triangle has adjacent = 12 and hypotenuse = 13. Find cos θ.
A right triangle has opposite = 8 and adjacent = 6. Find tan θ.
In a right triangle, sin θ = 5/13. Find cos θ.
In a right triangle, tan θ = 7/4. Find sec θ.
Section 3: Graphing & Transformations
Find the amplitude, period, phase shift, and vertical shift:
y = 3 sin(2x)
y = −2 cos(x/3) + 4
y = sin(x − π/2) − 1
y = 5 cos(4x + π)
y = −1/2 sin(x + π/3)
Section 4: Trigonometric Identities
Simplify each expression:
sin²x + cos²x
tan x ⋅ cot x
1 / sec x
sin x ⋅ cos x ⋅ (tan x)
(1 − cos²x) / sin x
Section 5: Verify Identities
Show whether the identity is true:
sec x − cos x = tan x ⋅ sin x
sin x / (1 + cos x) = (1 − cos x) / sin x
1 + tan²x = sec²x
sin(−x) = −sin x
cos(π − x) = −cos x
Section 6: Solve Trigonometric Equations
Solve for x on the interval 0 ≤ x < 2π:
sin x = 1/2
cos x = −√3 / 2
2 sin x = −1/2
tan x = 1
sin(2x) = sin x
Section 7: Word Problems
A Ferris wheel rotates in a sinusoidal pattern. Its height in feet is modeled by h(t) = 20 sin(π/6 t) + 30. Find the amplitude and vertical shift.
A sound wave is described by y = 4 cos(8x). Find the period.
A rotating lighthouse beam sweeps with function θ(t) = 2π t − π/4. Find the phase shift.
The depth of water at a pier is d(t) = 5 + 2 sin(π/12 t). Find the maximum depth.
Answer Key
Section 1: 1/2, −√2/2, −√3, 2, −2, 1
Section 2: 3/5, 12/13, 4/3, 12/13, √65 / 4
Section 3: A = 3, P = π, shift = none, V = 0; A = 2, P = 6π, shift = none, V = 4; A = 1, P = 2π, shift = +π/2 right, V = −1; A = 5, P = π/2, shift = −π/4, V = 0; A = 1/2, P = 2π, shift = −π/3, V = 0
Section 4: 1, 1, cos x, sin²x, sin x
Section 5: True, True, True, True, True
Section 6: x = π/6, 5π/6; x = 5π/6, 7π/6; x = 7π/6, 11π/6; x = π/4, 5π/4; x = 0, π
Section 7: Amplitude = 20; Vertical shift = 30; Period = π/4; Phase shift = +π/4; Maximum = 7