Trigonometric Functions (Introduction) — High School Algebra 2 practice questions

Evaluate cos(2π/3).

1. 1/2
2. √3/2
3. −√3/2
4. −1/2

Answer: D — −1/2

D) Correct — 2π/3 is in Quadrant II, where cosine is negative; the reference angle is π/3. A) Forgets the negative sign. B) Uses the sine value without the sign check. C) Uses the sine magnitude with a negative sign.

Which angle is coterminal with 400°?

1. 760°
2. −40°
3. 220°
4. 40°

Answer: D — 40°

D) Correct — Subtract 360°. A) Adds 360°. B) Changes the sign. C) Subtracts 180°.

What is tan 45°?

1. 1
2. √2/2
3. 0
4. Undefined

Answer: A — 1

A) Correct — sin 45° = cos 45°. B) Is sin 45°. C) Is tan 0°. D) Is tan 90°.

What is the reference angle for 210°?

1. 150°
2. 60°
3. 30°
4. 210°

Answer: C — 30°

C) Correct — 210° − 180°. A) Subtracts from 360°. B) Uses 270° − 210°. D) Doesn't reduce the angle.

If sin θ = 3/5 and cos θ = 4/5, what is tan θ?

1. 4/3
2. 3/4
3. 12/25
4. 7/5

Answer: B — 3/4

B) Correct — tan θ = sin θ / cos θ. A) Divides cosine by sine. C) Multiplies them. D) Adds them.

Convert 135° to radians.

1. 4π/3
2. 3π/4
3. 3π/2
4. 135π

Answer: B — 3π/4

B) Correct — 135 · π/180 = 3π/4. A) Inverts the fraction. C) Uses 270°. D) Forgets to divide by 180.

On the unit circle, the point at angle θ has coordinates:

1. (sin θ, cos θ)
2. (tan θ, 1)
3. (cos θ, sin θ)
4. (θ, 1)

Answer: C — (cos θ, sin θ)

C) Correct — x is cosine and y is sine. A) Swaps the coordinates. B) Is not a unit-circle point. D) Uses the angle as a coordinate.

What is the amplitude of y = 3 sin x?

1. 6
2. 2π
3. 3
4. 1

Answer: C — 3

C) Correct — The amplitude is |a|. A) Is the distance from maximum to minimum. B) Is the period. D) Is the amplitude of sin x.

Evaluate sin(7π/6).

1. 1/2
2. −√3/2
3. √3/2
4. −1/2

Answer: D — −1/2

D) Correct — 7π/6 is in Quadrant III, where sine is negative; the reference angle is π/6. A) Forgets the negative sign. B) Uses the cosine magnitude. C) Uses the cosine value.

In which quadrant is sin θ > 0 and cos θ < 0?

1. Quadrant I
2. Quadrant II
3. Quadrant III
4. Quadrant IV

Answer: B — Quadrant II

B) Correct — y is positive and x is negative there. A) Both are positive. C) Both are negative. D) Sine is negative there.

Worked Example: Solve 2 cos(x) + 1 = 0 on [0, 2π).

Step 1: Isolate cos(x). 2 cos(x) = −1 cos(x) = −1/2 Step 2: Find the reference angle. cos⁻¹(1/2) = π/3 (reference angle) Step 3: Determine quadrants where cosine is negative. Cosine is negative in quadrants II and III. Step 4: Calculate solutions in each quadrant. Q II: x = π − π/3 = 2π/3 Q III: x = π + π/3 = 4π/3 Step 5: Confirm both lie in [0, 2π). 2π/3 ≈ 2.094 ✓ ; 4π/3 ≈ 4.189 ✓ Final answer: x = 2π/3 and x = 4π/3.

How do you convert between degrees and radians?

Multiply degrees by π/180 to get radians; multiply radians by 180/π to get degrees. 180° = π radians.

Amplitude and period of y = a sin(bx)?

Amplitude = |a|. Period = 2π/|b|. Example: y = 3 sin(2x) has amplitude 3 and period π.

Which trig functions are positive in each quadrant?

QI: all. QII: sine. QIII: tangent. QIV: cosine. ("All Students Take Calculus.")