Master Series · Pre-Calculus

Trigonometry
Ultimate Quiz

All Units · Exam-Style · Multiple Choice
📐 20 Questions ⏱ 40 min 🎯 SAT / ACT Level 📊 Instant Score
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Unit 01

Angle Measures & Radian/Degree

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Conversion Formulas

Radians → Degrees: multiply by 180/π
Degrees → Radians: multiply by π/180
Arc length: s = rθ (θ in radians)
Area of sector: A = ½r²θ
π rad = 180° π/2 = 90° π/3 = 60° π/4 = 45° π/6 = 30°
Example
Convert 210° to radians.
210 × (π/180) = 7π/6
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Coterminal Angles

Angles that share the same terminal side. Add or subtract 360° (or 2π) to find coterminals.

θ ± 360°n (degrees)
θ ± 2πn (radians), n ∈ ℤ
Example
Find a positive coterminal angle of −50°.
−50° + 360° = 310°
Unit 02

Unit Circle & Trig Values

⭕

Unit Circle Coordinates

On the unit circle (r=1):
cos θ = x, sin θ = y

Key values:
0°: (1, 0) · 30°: (√3/2, 1/2)
45°: (√2/2, √2/2) · 60°: (1/2, √3/2)
90°: (0, 1) · 180°: (−1, 0)
270°: (0, −1)
Example
Find cos(5π/6).
Reference angle π/6, QII → cos = −√3/2
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Signs by Quadrant (CAST)

Q I: All positive
Q II: sin positive
Q III: tan positive
Q IV: cos positive
All Students Take Calculus
Example
If sin θ > 0 and cos θ < 0, which quadrant?
Quadrant II
Unit 03

Fundamental Identities

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Pythagorean Identities

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
Example
If cos θ = 3/5, find sin θ (θ in QI).
sin²θ = 1 − 9/25 = 16/25, so sin θ = 4/5
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Reciprocal & Quotient

csc θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ = cos θ/sin θ
tan θ = sin θ/cos θ
Example
Find sec(π/3).
1/cos(π/3) = 1/(1/2) = 2
Unit 04

Graphs of Trig Functions

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Amplitude, Period, Phase Shift

y = A sin(Bx − C) + D
• Amplitude: |A|
• Period: 2π/|B|
• Phase shift: C/B (right if positive)
• Vertical shift: D
Period of tan: π/|B|
Example
y = 3 sin(2x − π/4). Find amplitude, period, phase shift.
Amplitude = 3, Period = π, Phase shift = π/8 right
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Key Properties

sin x: odd, range [−1,1], period 2π
cos x: even, range [−1,1], period 2π
tan x: odd, range (−∞,∞), period π
Asymptotes of tan: x = π/2 + nπ
Example
Period of y = cos(πx/3)?
2π/(π/3) = 6
Unit 05

Sum, Difference & Double Angle

➕

Sum & Difference Formulas

sin(A±B) = sinA cosB ± cosA sinB
cos(A±B) = cosA cosB ∓ sinA sinB
tan(A±B) = (tanA ± tanB)/(1 ∓ tanA tanB)
Example
Exact value of sin(75°).
sin(45°+30°) = (√6+√2)/4
✖️

Double Angle Formulas

sin(2θ) = 2 sinθ cosθ
cos(2θ) = cos²θ − sin²θ
= 1 − 2sin²θ = 2cos²θ − 1
tan(2θ) = 2tanθ/(1 − tan²θ)
Example
If sin θ = 3/5 (QI), find sin(2θ).
2(3/5)(4/5) = 24/25
Unit 06

Inverse Trig Functions

↩️

Domains & Ranges

sin⁻¹x: domain [−1,1], range [−π/2, π/2]
cos⁻¹x: domain [−1,1], range [0, π]
tan⁻¹x: domain ℝ, range (−π/2, π/2)
Example
Find cos(sin⁻¹(5/13)).
Draw right triangle: adj = 12, so cos = 12/13
Unit 07

Law of Sines & Law of Cosines

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Law of Sines

a/sinA = b/sinB = c/sinC

Use when: AAS, ASA, SSA (ambiguous case)
Example
A=30°, B=60°, a=8. Find b.
8/sin30° = b/sin60° → b = 8√3
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Law of Cosines

a² = b² + c² − 2bc cosA
b² = a² + c² − 2ac cosB
c² = a² + b² − 2ab cosC

Use when: SAS, SSS
Example
a=5, b=7, C=60°. Find c.
c² = 25+49−2(5)(7)(1/2) = 39, c=√39
Exam Mode

20 Multiple Choice Questions

Select one answer per question. Explanations appear instantly after each answer.

0 / 20
01
Radian / Degree Conversion
Easy
Which of the following is the radian measure equivalent to 300°?
✦ Solution
300° × (π/180) = 300π/180 = 5π/3. Simplify: 300÷60=5, 180÷60=3. ∴ 5π/3. Answer: B
02
Arc Length
Medium
A circle has radius 6 cm. What is the arc length subtended by a central angle of 2π/3 radians?
✦ Solution
s = rθ = 6 × (2π/3) = 12π/3 = 4π cm. Answer: B
03
Unit Circle
Medium
What is the exact value of sin(4π/3)?
✦ Solution
4π/3 is in Quadrant III (π < 4π/3 < 3π/2). Reference angle = 4π/3 − π = π/3. sin(π/3) = √3/2. In QIII, sin is negative. ∴ sin(4π/3) = −√3/2. Answer: B
04
Pythagorean Identity
Medium
If sin θ = −5/13 and θ is in Quadrant III, what is cos θ?
✦ Solution
cos²θ = 1 − sin²θ = 1 − 25/169 = 144/169, so |cos θ| = 12/13. In QIII, cos is negative. ∴ cos θ = −12/13. Answer: B
05
Trig Identities
Medium
Which expression is equivalent to (1 − cos²θ)/sin θ?
✦ Solution
1 − cos²θ = sin²θ (Pythagorean identity). So (sin²θ)/sinθ = sin θ. Answer: B
06
Graphing – Period
Medium
What is the period of y = −2 cos(3x + π)?
✦ Solution
For y = A cos(Bx + C), the period is 2π/|B|. Here B = 3, so period = 2π/3. Answer: B
07
Graphing – Phase Shift
Medium
For y = sin(2x − π/2), what is the phase shift?
✦ Solution
Phase shift = C/B = (π/2)/2 = π/4. Because we write 2x − π/2 = 2(x − π/4), the shift is π/4 to the right. Answer: B
08
Sum Formula
Hard
What is the exact value of cos(15°)?
✦ Solution
cos(15°) = cos(45° − 30°) = cos45° cos30° + sin45° sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6+√2)/4. Answer: B
09
Double Angle
Hard
If cos θ = −3/5 and π/2 < θ < π, what is sin(2θ)?
✦ Solution
θ is in QII: sin θ > 0. sin²θ = 1 − 9/25 = 16/25, so sin θ = 4/5. sin(2θ) = 2 sin θ cos θ = 2(4/5)(−3/5) = −24/25. Answer: B
10
Inverse Trig
Medium
What is the exact value of tan(sin⁻¹(−1/2))?
✦ Solution
sin⁻¹(−1/2) = −π/6 (in range [−π/2, π/2]). tan(−π/6) = −tan(π/6) = −(1/√3) = −√3/3. Answer: A
11
Trig Equation
Medium
Solve 2 sin θ − 1 = 0 for 0 ≤ θ < 2π. Which pair of solutions is correct?
✦ Solution
2 sin θ = 1 → sin θ = 1/2. sin θ = 1/2 for θ = π/6 (QI) and θ = π − π/6 = 5π/6 (QII). Answer: A
12
Identity Simplification
Hard
Simplify: (sin x + cos x)² − 1
✦ Solution
(sin x + cos x)² = sin²x + 2 sin x cos x + cos²x = 1 + 2 sin x cos x. Subtract 1: result = 2 sin x cos x = sin(2x). Both A and B are equivalent, so C. Answer: C
13
Law of Sines
Medium
In triangle ABC, A = 45°, B = 75°, and a = 10. What is b (to the nearest tenth)?
✦ Solution
a/sinA = b/sinB → 10/sin45° = b/sin75°. b = 10 × sin75°/sin45° = 10 × 0.9659/0.7071 ≈ 10 × 1.366 ≈ 13.7. Answer: A
14
Law of Cosines
Hard
In a triangle with sides a = 7, b = 8, c = 9, what is cos A?
✦ Solution
Law of Cosines: a² = b² + c² − 2bc cosA → 49 = 64 + 81 − 144 cosA → 144 cosA = 96 → cosA = 96/144 = 2/3. Answer: B
15
Amplitude & Midline
Medium
A sinusoidal function has a maximum value of 7 and minimum value of −3. What is the amplitude?
✦ Solution
Amplitude = (max − min)/2 = (7 − (−3))/2 = 10/2 = 5. Midline = (7 + (−3))/2 = 2. Answer: B
16
Half-Angle Formula
Hard
Using the half-angle formula, which expression equals sin(π/8)?
✦ Solution
Half-angle: sin(θ/2) = √((1 − cosθ)/2). Here θ/2 = π/8 means θ = π/4. cos(π/4) = √2/2. So sin(π/8) = √((1 − √2/2)/2). Since π/8 is in QI, we take positive root. Answer: B
17
Coterminal / Reference Angle
Easy
What is the reference angle for θ = 7π/6?
✦ Solution
7π/6 is in QIII (π < 7π/6 < 3π/2). Reference angle = θ − π = 7π/6 − π = π/6. Answer: A
18
Trig Equation – Multiple Angles
Hard
How many solutions does cos(2x) = 1/2 have on [0, 2π)?
✦ Solution
Let u = 2x. Since x ∈ [0, 2π), u ∈ [0, 4π). cos u = 1/2 → u = π/3, 5π/3, π/3+2π=7π/3, 5π/3+2π=11π/3. That gives 4 solutions: x = π/6, 5π/6, 7π/6, 11π/6. Answer: B
19
Area of Triangle
Medium
Find the area of a triangle with sides b = 5, c = 8, and included angle A = 30°.
✦ Solution
Area = (1/2) b c sin A = (1/2)(5)(8) sin 30° = 20 × (1/2) = 10. Answer: A
20
Composite Inverse Trig
Hard
What is the exact value of cos(tan⁻¹(2))?
✦ Solution
Let θ = tan⁻¹(2): tan θ = 2 = opp/adj = 2/1. Hypotenuse = √(4+1) = √5. cos θ = adj/hyp = 1/√5 = √5/5. Answer: A
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