Pre-Calculus Trigonometry | 20 Core Problems

C-01 Angle Measurement & Conversion

Angles are measured in degrees or radians. A full rotation = 360° = 2π radians.

⚡ MEMORIZE

Degrees → Radians: multiply by π/180
Radians → Degrees: multiply by 180/π
Common: 30°=π/6 · 45°=π/4 · 60°=π/3 · 90°=π/2
120°=2π/3 · 135°=3π/4 · 150°=5π/6 · 180°=π

EXAMPLE

Convert 210° to radians.

210 × (π/180) = 7π/6

C-02 Unit Circle & Trig Values

On the unit circle (radius = 1), for angle θ: cos θ = x-coordinate, sin θ = y-coordinate, tan θ = y/x.

⚡ MEMORIZE — KEY VALUES

sin(0)=0 sin(π/6)=1/2 sin(π/4)=√2/2
sin(π/3)=√3/2 sin(π/2)=1
cos(0)=1 cos(π/6)=√3/2 cos(π/4)=√2/2
cos(π/3)=1/2 cos(π/2)=0
tan(π/4)=1 tan(π/6)=√3/3 tan(π/3)=√3

EXAMPLE

Find sin(5π/6).

5π/6 is in Q2, reference angle = π/6. sin(5π/6) = sin(π/6) = 1/2

C-03 Pythagorean & Reciprocal Identities

⚡ MEMORIZE

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
csc θ = 1/sinθ · sec θ = 1/cosθ · cot θ = cosθ/sinθ

EXAMPLE

If sin θ = 3/5, find cos θ (θ in Q1).

cos²θ = 1 − (9/25) = 16/25 → cos θ = 4/5

C-04 Graphs of Sine & Cosine

General form: y = A sin(Bx − C) + D

⚡ MEMORIZE

Amplitude = |A|
Period = 2π / |B|
Phase shift = C / B (right if positive)
Vertical shift = D
Range = [D−|A|, D+|A|]

EXAMPLE

y = 3 sin(2x − π) + 1. Find amplitude, period, phase shift.

Amplitude = 3 · Period = 2π/2 = π · Phase shift = π/2 (right)

C-05 Angle Addition & Double Angle Formulas

⚡ MEMORIZE

sin(A±B) = sinA cosB ± cosA sinB
cos(A±B) = cosA cosB ∓ sinA sinB
sin(2A) = 2 sinA cosA
cos(2A) = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1
tan(A+B) = (tanA + tanB)/(1 − tanA tanB)

EXAMPLE

Find sin(75°) = sin(45° + 30°).

= sin45°cos30° + cos45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6+√2)/4

C-06 Inverse Trig Functions

Inverse trig functions return the angle whose trig value equals x.

⚡ MEMORIZE — DOMAINS & RANGES

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

EXAMPLE

Find arcsin(−1/2).

We need θ ∈ [−π/2, π/2] with sin θ = −1/2. Answer: −π/6

C-07 Law of Sines & Law of Cosines

⚡ MEMORIZE

Law of Sines: a/sinA = b/sinB = c/sinC
Law of Cosines: c² = a² + b² − 2ab cosC
Area = (1/2) ab sinC

EXAMPLE

Triangle: a=7, b=10, C=60°. Find c.

c² = 49 + 100 − 2(7)(10)cos60° = 149 − 70 = 79 → c = √79

Trig Master Quiz

Concept Review

Practice Problems

Answer Key & Solutions