📐 Concept Review & Memory Guide
Unit 1 · Angle Measures & Radian-Degree Conversion
Radian: arc length / radius. Full circle = 2π radians = 360°.
Conversion: deg × π/180 = rad | rad × 180/π = deg
Conversion: deg × π/180 = rad | rad × 180/π = deg
180° = π radone half circle
90° = π/2 radone quarter circle
45° = π/4 radeighth circle
30° = π/6 radtwelfth circle
Trick "Multiply by π/180 to go to radians; think π-over-half-circle."
Unit 2 · Unit Circle & Reference Angles
On the unit circle (radius = 1), the point at angle θ is (cos θ, sin θ).
Key values to memorize:
Key values to memorize:
sin 30° = 1/2cos 30° = √3/2
sin 45° = √2/2cos 45° = √2/2
sin 60° = √3/2cos 60° = 1/2
sin 90° = 1cos 90° = 0
Mnemonic sin values 0°→90°: √0/2, √1/2, √2/2, √3/2, √4/2 = 0, 1/2, √2/2, √3/2, 1
Unit 3 · Core Identities
sin²θ + cos²θ = 1Pythagorean #1
1 + tan²θ = sec²θPythagorean #2
1 + cot²θ = csc²θPythagorean #3
tan θ = sin θ / cos θQuotient
sin(−θ) = −sin θOdd function
cos(−θ) = cos θEven function
Unit 4 · Sum, Difference & Double Angle Formulas
sin(A±B) = sinAcosB ± cosAsinBSum / Difference
cos(A±B) = cosAcosB ∓ sinAsinBSum / Difference
sin 2θ = 2 sin θ cos θDouble Angle
cos 2θ = cos²θ − sin²θDouble Angle
cos 2θ = 2cos²θ − 1Double Angle alt
cos 2θ = 1 − 2sin²θDouble Angle alt
Unit 5 · Graphs: Amplitude, Period, Phase Shift
For y = A sin(Bx + C) + D and y = A cos(Bx + C) + D:
Amplitude = |A| | Period = 2π/|B| | Phase shift = −C/B | Vertical shift = D
Amplitude = |A| | Period = 2π/|B| | Phase shift = −C/B | Vertical shift = D
Memory "ABCD: Amplitude, B-determines period (2π/B), C shifts left/right (−C/B), D shifts up/down."
Unit 6 · Laws of Sines & Cosines
a/sinA = b/sinB = c/sinCLaw of Sines
c² = a² + b² − 2ab cosCLaw of Cosines
Area = ½ ab sinCTriangle Area
Unit 7 · Inverse Trig & ASTC Quadrant Signs
ASTC (All Students Take Calculus):
Q1: All positive | Q2: Sin positive | Q3: Tan positive | Q4: Cos positive
Q1: All positive | Q2: Sin positive | Q3: Tan positive | Q4: Cos positive
arcsin: [−π/2, π/2]Domain [−1,1]
arccos: [0, π]Domain [−1,1]
arctan: (−π/2, π/2)Domain all reals
Example
Find sin(5π/6).
5π/6 is in Q2 where sin > 0. Reference angle = π − 5π/6 = π/6. sin(π/6) = 1/2. ∴ sin(5π/6) = 1/2 ✓
5π/6 is in Q2 where sin > 0. Reference angle = π − 5π/6 = π/6. sin(π/6) = 1/2. ∴ sin(5π/6) = 1/2 ✓
Example
Simplify (1 − cos 2θ) / 2.
cos 2θ = 1 − 2sin²θ, so 1 − cos 2θ = 2sin²θ. Divide by 2: sin²θ ✓
cos 2θ = 1 − 2sin²θ, so 1 − cos 2θ = 2sin²θ. Divide by 2: sin²θ ✓
📝 Practice Problems
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