Study Guide
Core Concepts & Formulas to Master
① Angle Measurement: Degrees & Radians
Angles are measured in degrees or radians. Converting between them is a fundamental skill.
Radians = Degrees × (π / 180)
Degrees = Radians × (180 / π)
Arc Length: s = rθ (θ must be in radians)
Sector Area: A = ½ r²θ
180° = π
90° = π/2
60° = π/3
45° = π/4
30° = π/6
② Unit Circle — Key Values
On the unit circle (radius = 1): the point at angle θ is (cos θ, sin θ). Memorize all four quadrants.
θ = 0: (1, 0) sin=0, cos=1, tan=0
θ = π/6: (√3/2, ½) sin=½, cos=√3/2, tan=1/√3
θ = π/4: (√2/2, √2/2) sin=√2/2, cos=√2/2, tan=1
θ = π/3: (½, √3/2) sin=√3/2, cos=½, tan=√3
θ = π/2: (0, 1) sin=1, cos=0, tan=undef
🔑 ASTC: All → Sin → Tan → Cos (Q1→Q2→Q3→Q4 positive)
sin(−θ) = −sinθ (odd)
cos(−θ) = cosθ (even)
③ Trig Graphs: Amplitude, Period, Phase Shift
Standard form: y = A sin(Bx + C) + D or y = A cos(Bx + C) + D
Amplitude = |A|
Period = 2π / |B|
Phase Shift = −C / B (negative = left shift)
Vertical Shift = D
Period of tan: π (not 2π)
Amplitude of tan/cot: undefined
④ Pythagorean & Fundamental Identities
sin²θ + cos²θ = 1 → sin²θ = 1−cos²θ, cos²θ = 1−sin²θ
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
Reciprocals: csc=1/sin, sec=1/cos, cot=1/tan
Quotient: tan=sin/cos, cot=cos/sin
sec²−1 = tan²
csc²−1 = cot²
⑤ Sum & Difference Formulas
sin(A ± B) = sinA cosB ± cosA sinB
cos(A ± B) = cosA cosB ∓ sinA sinB
tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)
sin: same sign ±
cos: OPPOSITE sign ∓
⑥ Double & Half Angle Formulas
sin 2θ = 2 sinθ cosθ
cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1
tan 2θ = 2tanθ / (1 − tan²θ)
Half Angle:
sin(θ/2) = ±√[(1 − cosθ)/2]
cos(θ/2) = ±√[(1 + cosθ)/2]
3 forms of cos 2θ — choose the most useful
cos2θ = 1−2sin²θ (eliminate cos)
cos2θ = 2cos²θ−1 (eliminate sin)
⑦ Law of Sines & Law of Cosines
Law of Sines: a / sinA = b / sinB = c / sinC
Law of Cosines: c² = a² + b² − 2ab cosC
Rearranged: cosC = (a²+b²−c²) / (2ab)
LoS: use for AAS, ASA, SSA
LoC: use for SAS, SSS
⚠ SSA Ambiguous Case: h=b·sinA; if h<a<b → 2 triangles
⑧ Inverse Trig Functions — Domain & Range
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)
arcsin & arctan: [−π/2, π/2]
arccos: [0, π] ← different!
Evaluate by right triangle or unit circle
Warm-Up
Worked Examples
Example 1 — Radian Conversion
Q: Convert 240° to radians.
240 × (π/180) = 240π/180 = 4π/3
Answer: 4π/3
Example 2 — Pythagorean Identity
Q: If sin θ = 3/5 and θ ∈ Q1, find cos θ.
cos²θ = 1 − (3/5)² = 1 − 9/25 = 16/25 → cosθ = 4/5
Answer: 4/5
Example 3 — Double Angle
Q: If cos θ = −1/2 and θ ∈ Q2, find sin 2θ.
sinθ = √3/2 (Q2, positive)
sin 2θ = 2(√3/2)(−1/2) = −√3/2
Answer: −√3/2