Trigonometry — Cheat Sheet

THE 6 FUNCTIONS

SOH-CAH-TOA (right triangles)
Sin=Opp/Hyp · Cos=Adj/Hyp · Tan=Opp/Adj

sin θ

opp/hyp = y (unit circle)

cos θ

adj/hyp = x (unit circle)

tan θ

opp/adj = sinθ/cosθ

csc θ

1/sinθ = hyp/opp

sec θ

1/cosθ = hyp/adj

cot θ

1/tanθ = cosθ/sinθ

CAST RULE

All Students Take Calculus

Q II · 90°–180°

sin + · cos − · tan −

Q I · 0°–90°

ALL positive

Q III · 180°–270°

sin − · cos − · tan +

Q IV · 270°–360°

sin − · cos + · tan −

CONVERSIONS

Deg → Rad

× π/180

Rad → Deg

× 180/π

Arc length

s = r·θ (θ in rad!)

Ref angle QII

180° − θ

Ref angle QIII

θ − 180°

Ref angle QIV

360° − θ

KEY UNIT CIRCLE VALUES

θ°	θ rad	sin θ	cos θ	tan θ
0°	0	0	1	0
30°	π/6	1/2	√3/2	√3/3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	—
120°	2π/3	√3/2	−1/2	−√3
135°	3π/4	√2/2	−√2/2	−1
150°	5π/6	1/2	−√3/2	−√3/3
180°	π	0	−1	0
210°	7π/6	−1/2	−√3/2	√3/3
225°	5π/4	−√2/2	−√2/2	1
240°	4π/3	−√3/2	−1/2	√3
270°	3π/2	−1	0	—
300°	5π/3	−√3/2	1/2	−√3
315°	7π/4	−√2/2	√2/2	−1
330°	11π/6	−1/2	√3/2	−√3/3
360°	2π	0	1	0

SINUSOIDAL FUNCTION

f(x) = a · sin( b(x − h) ) + k

a — amplitude

|a|

period

2π / |b|

frequency

|b| / (2π)

h — phase shift

+h = right, −h = left

k — vert. shift

midline at y = k

maximum

k + |a|

minimum

k − |a|

PYTHAGOREAN IDENTITIES

Primary ★

sin²θ + cos²θ = 1

÷ cos²θ

tan²θ + 1 = sec²θ

÷ sin²θ

1 + cot²θ = csc²θ

RECIPROCAL IDENTITIES

csc θ

1/sinθ

sec θ

1/cosθ

cot θ

1/tanθ

tan θ

sinθ / cosθ

PERIODICITY

sin(−θ) = −sin θ [odd]
cos(−θ) = cos θ [even]
sin(θ+2πn) = sinθ, cos(θ+2πn) = cosθ
tan(θ+πn) = tanθ

DOUBLE ANGLE

sin(2θ)

2 sinθ cosθ

cos(2θ)

cos²θ − sin²θ

2cos²θ − 1

1 − 2sin²θ

tan(2θ)

2tanθ / (1−tan²θ)

SUM & DIFFERENCE

sin(α±β)

sinα cosβ ± cosα sinβ

cos(α±β)

cosα cosβ ∓ sinα sinβ

tan(α±β)

(tanα±tanβ)/(1∓tanα tanβ)

INVERSE TRIG

sin⁻¹(x)

domain [−1,1] · range [−π/2, π/2]

cos⁻¹(x)

domain [−1,1] · range [0, π]

tan⁻¹(x)

domain ℝ · range (−π/2, π/2)

⚠ Always check for a 2nd solution using the CAST rule!

LAW OF SINES & COSINES

Sines

sinA/a = sinB/b = sinC/c

Cosines

a² = b²+c² − 2bc·cosA

Sines: use for AAS/ASA/SSA · Cosines: use for SSS/SAS

PROVING IDENTITIES

Work ONE side only — never cross the =
Convert to sin & cos if stuck
Look for Pythagorean substitutions
Try factoring (diff. of squares)
Multiply by conjugate if needed