Pre-Calculus Trigonometry Master Quiz

Unit 1

Angles & Radian Measure

Key Formulas to Memorize

Degrees → Radians: θ(rad) = θ(deg) × π/180 Radians → Degrees: θ(deg) = θ(rad) × 180/π Arc Length: s = rθ (θ in radians) Sector Area: A = ½r²θ (θ in radians) Angular velocity: ω = θ/t, Linear velocity: v = rω

RadiansArc LengthSector Area

Example

Convert 150° to radians.
150 × π/180 = 5π/6

Unit 2

The Unit Circle

Critical Values — Memorize This Table

θ (deg)	θ (rad)	sin θ	cos θ	tan θ
0°	0	0	1	0
30°	π/6	1/2	√3/2	1/√3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	undef

ASTC RuleQuadrants

ASTC Memory Aid

All Students Take Calculus
Q1: All positive · Q2: Sin positive · Q3: Tan positive · Q4: Cos positive

Unit 3

Trigonometric Identities

Fundamental Identities

Pythagorean: sin²θ + cos²θ = 1 1 + tan²θ = sec²θ 1 + cot²θ = csc²θ Reciprocal: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ Quotient: tan θ = sin θ/cos θ, cot θ = cos θ/sin θ Cofunction: sin θ = cos(90°−θ), tan θ = cot(90°−θ)

PythagoreanReciprocalQuotientCofunction

Unit 4

Sum, Difference & Double-Angle Formulas

Essential Formulas

Sum/Difference: sin(A±B) = sinA cosB ± cosA sinB cos(A±B) = cosA cosB ∓ sinA sinB tan(A±B) = (tanA ± tanB)/(1 ∓ tanA tanB) Double-Angle: sin 2θ = 2 sinθ cosθ cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ tan 2θ = 2tanθ/(1 − tan²θ) Half-Angle: sin(θ/2) = ±√((1−cosθ)/2) cos(θ/2) = ±√((1+cosθ)/2)

Example

Find sin 75°.
sin(45°+30°) = sin45°cos30° + cos45°sin30°
= (√2/2)(√3/2) + (√2/2)(1/2) = (√6+√2)/4

Unit 5

Graphs of Trig Functions

y = A sin(Bx + C) + D

Amplitude: |A| Period: 2π/|B| (for sin, cos) or π/|B| (for tan, cot) Phase Shift: −C/B (left if C/B > 0; right if C/B < 0) Vertical Shift: D

AmplitudePeriodPhase Shift

Example

For y = 3 sin(2x − π/2) + 1:
Amplitude = 3, Period = π, Phase shift = π/4 right, Vertical shift = up 1

Unit 6

Inverse Trigonometric Functions

Domains & Ranges

arcsin(x): domain [−1,1], range [−π/2, π/2] arccos(x): domain [−1,1], range [0, π] arctan(x): domain (−∞,∞), range (−π/2, π/2)

Example

Find arctan(√3).
tan(π/3) = √3, and π/3 is in (−π/2, π/2), so arctan(√3) = π/3

Unit 7

Law of Sines & Law of Cosines

Solving Triangles

Law of Sines: a/sin A = b/sin B = c/sin C Law of Cosines: c² = a² + b² − 2ab cos C cos C = (a² + b² − c²)/(2ab) Area of Triangle: Area = ½ ab sin C Area = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2 [Heron's]

ASA/AAS → Law of SinesSSS/SAS → Law of Cosines

Unit 8

Polar Coordinates & Complex Numbers

Key Conversions

Rectangular ↔ Polar: x = r cosθ, y = r sinθ r = √(x²+y²), θ = arctan(y/x) Complex (Polar Form): z = r(cosθ + i sinθ) = re^(iθ) De Moivre's Theorem: zⁿ = rⁿ(cos nθ + i sin nθ)

Pre-Calculus Trigonometry

Key Formulas to Memorize

Critical Values — Memorize This Table

Fundamental Identities

Essential Formulas

y = A sin(Bx + C) + D

Domains & Ranges

Solving Triangles

Key Conversions

🎯 Exam Simulation

📋 Full Answer Key & Explanations