Concepts & Formulas
Unit 1
Angles & Radian Measure
Degrees ↔ Radians conversion is fundamental. One full rotation = 360° = 2π radians.
θ(rad) = θ(deg) × π/180
θ(deg) = θ(rad) × 180/π
Arc length s = rθ (θ must be in radians)
Sector area A = ½r²θ (θ must be in radians)
Key conversions to memorize: 30°=π/6, 45°=π/4, 60°=π/3, 90°=π/2, 120°=2π/3, 150°=5π/6, 180°=π, 210°=7π/6, 240°=4π/3, 270°=3π/2, 300°=5π/3, 330°=11π/6, 360°=2π
Example: Convert 240° to radians.
240 × (π/180) = 240π/180 = 4π/3
Answer: 4π/3 radians ≈ 4.189 rad
Unit 2
Unit Circle & Standard Values
The unit circle (radius = 1, center at origin). For angle θ, the point is (cos θ, sin θ).
ASTC Rule (quadrant signs): All Pos | Sin Pos | Tan Pos | Cos Pos (QI→QII→QIII→QIV)
| Angle | 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 | undef |
| 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 |
| 240° | 4π/3 | −√3/2 | −1/2 | √3 |
| 270° | 3π/2 | −1 | 0 | undef |
| 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 |
Unit 3
Six Trig Functions & Reciprocals
sin θ = opp/hyp csc θ = hyp/opp = 1/sin θ
cos θ = adj/hyp sec θ = hyp/adj = 1/cos θ
tan θ = opp/adj cot θ = adj/opp = 1/tan θ
tan θ = sin θ/cos θ cot θ = cos θ/sin θ
Example: Right triangle with legs 5, 12 and hypotenuse 13. Find all 6 ratios for θ (opposite side = 5).
sin θ = 5/13 cos θ = 12/13 tan θ = 5/12
csc θ = 13/5 sec θ = 13/12 cot θ = 12/5
Unit 4
Pythagorean Identities
sin²θ + cos²θ = 1 ← MASTER IDENTITY
1 + tan²θ = sec²θ ← divide master by cos²θ
cot²θ + 1 = csc²θ ← divide master by sin²θ
Derived forms: sin²θ = 1 − cos²θ, tan²θ = sec²θ − 1, cot²θ = csc²θ − 1
Example: If sin θ = 3/5 and θ is in QII, find cos θ.
cos²θ = 1 − 9/25 = 16/25 → cos θ = ±4/5
QII: cos is negative → cos θ = −4/5
Unit 5
Graphs: Amplitude, Period, Phase Shift
y = A·sin(Bx − C) + D or y = A·cos(Bx − C) + D
Amplitude = |A|
Period = 2π / |B|
Phase shift = C/B (+ = right, − = left)
Midline = y = D
y = A·tan(Bx − C) + D
Period of tan/cot = π / |B|
Example: y = −2cos(3x − π) + 4
A=−2, B=3, C=π, D=4
Amplitude = 2 | Period = 2π/3
Phase shift = π/3 (right) | Midline: y = 4
Unit 6
Sum & Difference Identities
sin(A ± B) = sinA cosB ± cosA sinB
cos(A ± B) = cosA cosB ∓ sinA sinB
tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA·tanB)
Example: Find exact value of sin(75°).
sin(45°+30°) = sin45°cos30° + cos45°sin30°
= (√2/2)(√3/2) + (√2/2)(1/2)
= √6/4 + √2/4 = (√6 + √2)/4
Unit 7
Double & Half Angle Formulas
sin(2θ) = 2 sinθ cosθ
cos(2θ) = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1
tan(2θ) = 2tanθ / (1 − tan²θ)
Power-Reducing:
sin²θ = (1 − cos2θ)/2
cos²θ = (1 + cos2θ)/2
Half-Angle (sign depends on quadrant of θ/2):
sin(θ/2) = ±√((1 − cosθ)/2)
cos(θ/2) = ±√((1 + cosθ)/2)
Example: sin θ = 4/5 (QI). Find sin(2θ) and cos(2θ).
cos θ = 3/5 (QI, positive)
sin(2θ) = 2·(4/5)·(3/5) = 24/25
cos(2θ) = 1 − 2sin²θ = 1 − 32/25 = −7/25
Unit 8
Inverse Trig Functions
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)
Ranges are restricted so each inverse is a function. arcsin/arctan: QI or QIV. arccos: QI or QII.
Example: Find arctan(√3).
tan(θ) = √3 and θ ∈ (−π/2, π/2)
→ θ = π/3 (since tan(π/3) = √3 ✓)
arctan(√3) = π/3
Unit 9
Trig Equations
Step 1: Isolate the trig function. Step 2: Find reference angle. Step 3: Apply ASTC for all solutions in given interval.
General solutions (all real θ):
sin θ = k → θ = arcsin(k) + 2πn or π−arcsin(k) + 2πn
cos θ = k → θ = ±arccos(k) + 2πn
tan θ = k → θ = arctan(k) + πn
Example: Solve 2sin²x − sinx − 1 = 0 on [0°, 360°).
Factor: (2sinx + 1)(sinx − 1) = 0
sinx = −1/2 → x = 210°, 330°
sinx = 1 → x = 90°
Solution set: {90°, 210°, 330°}
Unit 10
Law of Sines & Law of Cosines
Law of Sines:
a/sinA = b/sinB = c/sinC
Law of Cosines:
c² = a² + b² − 2ab cosC
b² = a² + c² − 2ac cosB
a² = b² + c² − 2bc cosA
Triangle Area: K = ½ab sinC
Use Law of Sines for: AAS, ASA, SSA (ambiguous). Use Law of Cosines for: SAS, SSS.
Example: a=7, b=5, C=60°. Find c.
c² = 49 + 25 − 2(7)(5)cos60°
= 74 − 70(0.5) = 74 − 35 = 39
c = √39 ≈ 6.24