Unit 01 · Angle Measurement
Degree ↔ Radian Conversion
Degrees → Radians : × (π / 180)
Radians → Degrees : × (180 / π)
Full circle : 360° = 2π rad
Half circle : 180° = π rad
90° = π/2 60° = π/3
45° = π/4 30° = π/6
Example
Convert 120° to radians.
120 × π/180 = 2π/3
Unit 02 · Unit Circle
Key Angle Coordinates (cos θ, sin θ)
0° : (1, 0) 90° : (0, 1)
180° : (−1, 0) 270° : (0, −1)
30° : (√3/2, 1/2) 60° : (1/2, √3/2)
45° : (√2/2, √2/2)
ASTC rule — Q I: all +, Q II: sin +,
Q III: tan +, Q IV: cos +
Example
sin(210°) and cos(210°) ?
Q III, ref=30°: sin=−1/2, cos=−√3/2
Unit 03 · Right Triangle
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
Example
sin θ = 3/5 in Q I → find tan θ
adj = 4 → tan θ = 3/4
Unit 04 · Pythagorean Identities
The Three Fundamental Identities
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
Even/Odd:
cos(−θ) = cos θ (even)
sin(−θ) = −sin θ (odd)
tan(−θ) = −tan θ (odd)
Example
cos θ = −2/3, Q III → sin θ ?
sin θ = −√5/3 (negative in Q III)
Unit 05 · Graph Transformations
y = A·sin(Bx + C) + D
Amplitude : |A|
Period : 2π / |B|
Phase shift : −C/B
Vertical shift: D
Range : [D−|A|, D+|A|]
Example
y = 3sin(2x − π) + 1
Amp=3, Period=π, Shift=π/2 right, D=1
Unit 06 · Tangent Graph
y = A·tan(Bx + C)
Period : π / |B|
Asymptotes when: Bx+C = π/2 + nπ
x-intercepts when: Bx+C = nπ
No amplitude (unbounded range)
Example
y = tan(2x): period and asymptotes?
Period = π/2; asymptotes: x = π/4 + nπ/2
Unit 07 · Sum & Difference
Addition Formulas
sin(A ± B) = sinA cosB ± cosA sinB
cos(A ± B) = cosA cosB ∓ sinA sinB
tan(A ± B) = (tanA ± tanB)
÷ (1 ∓ tanA tanB)
Example
sin(75°) = sin(45° + 30°)
= (√6 + √2) / 4
Unit 08 · Double & Half Angle
Power-Reduction Formulas
sin(2θ) = 2 sinθ cosθ
cos(2θ) = cos²θ − sin²θ
= 2cos²θ − 1 = 1 − 2sin²θ
tan(2θ) = 2tanθ / (1 − tan²θ)
sin(θ/2) = ±√[(1 − cosθ)/2]
cos(θ/2) = ±√[(1 + cosθ)/2]
Example
sin θ = 3/5, Q I → sin(2θ)?
cos θ = 4/5 → sin(2θ) = 2(3/5)(4/5) = 24/25
Unit 09 · Inverse Trig
Inverse Functions & Restricted Ranges
arcsin : range [−π/2, π/2]
arccos : range [0, π]
arctan : range (−π/2, π/2)
sin(arcsin x) = x, x ∈ [−1,1]
arcsin(sin θ) = θ only if θ∈[−π/2,π/2]
Example
arccos(−1/2) = ?
cos(2π/3) = −1/2, so arccos(−1/2) = 2π/3
Unit 10 · Equations & Laws
Solving Trig Equations + Law of Sines/Cosines
For sin θ = k (general):
θ = arcsin(k) + 2nπ OR π−arcsin(k) + 2nπ
Law of Sines : a/sinA = b/sinB = c/sinC
Law of Cosines: c² = a²+b²−2ab cosC
Example
Solve 2sinθ − 1 = 0 on [0, 2π)
sinθ = 1/2 → θ = π/6, 5π/6