Pre-Calculus Trigonometry Mastery Test

Unit 1

Angles, Degrees & Radians

Angles are measured in degrees (°) or radians (rad). A full rotation = 360° = 2π rad. To convert, use the bridge factor π/180.

Conversion Rules

Degrees → Radians: multiply by π/180
Radians → Degrees: multiply by 180/π

30° = π/6 | 45° = π/4 | 60° = π/3
90° = π/2 | 120° = 2π/3 | 180° = π
270° = 3π/2 | 315° = 7π/4 | 360° = 2π

Worked Example

Convert 150° to radians:
150 × (π/180) = 150π/180 = 5π/6

Unit 2

The Unit Circle

A circle of radius 1 centered at the origin. Any point at angle θ on the circle has coordinates (cos θ, sin θ).

Special Angles — Commit These to Memory

θ = 0: (1, 0)
θ = π/6: (√3/2, 1/2)
θ = π/4: (√2/2, √2/2)
θ = π/3: (1/2, √3/2)
θ = π/2: (0, 1)
θ = 2π/3: (−1/2, √3/2)
θ = π: (−1, 0)
θ = 7π/6: (−√3/2, −1/2)
θ = 3π/2: (0, −1)

Worked Example

Find sin(5π/6). 5π/6 is in QII; reference angle = π − 5π/6 = π/6. In QII sine is positive.
sin(5π/6) = sin(π/6) = 1/2

Unit 3

The Six Trigonometric Functions

Defined by a right triangle: opposite (opp), adjacent (adj), hypotenuse (hyp).

SOH-CAH-TOA + Reciprocals

sin θ = opp/hyp     csc θ = hyp/opp
cos θ = adj/hyp     sec θ = hyp/adj
tan θ = opp/adj     cot θ = adj/opp

tan θ = sin θ / cos θ     cot θ = cos θ / sin θ

Worked Example

Right triangle: opp = 3, adj = 4, hyp = 5 (verify: 3²+4²=25=5² ✓)
sin θ = 3/5 | cos θ = 4/5 | tan θ = 3/4 | sec θ = 5/4

Unit 4

Reference Angles & Signs by Quadrant

The reference angle is the acute angle (0° to 90°) between the terminal side and the x-axis.

ASTC Rule — Signs by Quadrant

QI: sin+, cos+, tan+ "All positive"
QII: sin+, cos−, tan− "Sin positive"
QIII: sin−, cos−, tan+ "Tan positive"
QIV: sin−, cos+, tan− "Cos positive"

Mnemonic: "All Students Take Calculus"

Reference angle: QII: 180°−θ | QIII: θ−180° | QIV: 360°−θ

Unit 5

Pythagorean & Fundamental Identities

These are the foundation of every trig proof and simplification.

Core Identities to Memorize

sin²θ + cos²θ = 1
tan²θ + 1 = sec²θ
1 + cot²θ = csc²θ

sin(−θ) = −sin θ (odd function)
cos(−θ) = cos θ (even function)
tan(−θ) = −tan θ (odd function)

Unit 6

Graphs of Sine & Cosine

Standard form: y = A sin(Bx − C) + D or y = A cos(Bx − C) + D

Parameter Meanings

Amplitude = |A|
Period = 2π / |B|
Phase shift = C/B (right if C/B > 0)
Vertical shift = D

y = sin x: period 2π, passes through (0, 0)
y = cos x: period 2π, passes through (0, 1)

Worked Example

y = 3 sin(2x − π) + 1
A=3 → amplitude = 3 | B=2 → period = π
Phase shift = π/2 right | Vertical shift = 1 up

Unit 7

Sum & Difference Formulas

Formulas

sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

Worked Example

cos(75°) = cos(45° + 30°)
= cos45° cos30° − sin45° sin30°
= (√2/2)(√3/2) − (√2/2)(1/2) = (√6 − √2)/4

Unit 8

Double & Half-Angle Formulas

Double-Angle (memorize all three forms of cos 2θ)

sin(2θ) = 2 sin θ cos θ
cos(2θ) = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
tan(2θ) = 2 tan θ / (1 − tan²θ)

Half-Angle:
sin(θ/2) = ±√((1 − cos θ)/2)
cos(θ/2) = ±√((1 + cos θ)/2)

Unit 9

Inverse Trigonometric Functions

Inverse trig functions return the angle whose trig value equals x. The range is restricted to make them 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)

Worked Example

arctan(√3) = ? Find θ in (−π/2, π/2) where tan θ = √3.
tan(π/3) = √3 → arctan(√3) = π/3

Unit 10

Law of Sines & Law of Cosines

Used to solve any triangle when you know enough information (not just right triangles).

Law of Sines

a / sin A = b / sin B = c / sin C

Law of Cosines:
a² = b² + c² − 2bc cos A
b² = a² + c² − 2ac cos B
c² = a² + b² − 2ab cos C

Worked Example (Law of Cosines)

a = 5, b = 7, C = 60°. Find c:
c² = 25 + 49 − 2(5)(7)(0.5) = 74 − 35 = 39
c = √39 ≈ 6.24

Trigonometry
Mastery Test

Trigonometry — 20 Questions

Well done!

Pre-Calculus Trigonometry — 20-Question Mastery Test

Answer Key & Solutions

TrigonometryMastery Test

Trigonometry — 20 Questions

Well done!

Pre-Calculus Trigonometry — 20-Question Mastery Test

Answer Key & Solutions

Trigonometry
Mastery Test