Topic A: Right Triangle Trig & Radian Measure
Lessons 1–5 · SOH-CAH-TOA review, converting degrees/radians.
01–03 — Lessons 1–3
Right Triangle Trigonometry
- SOH-CAH-TOA:
- sin(θ) = opposite/hypotenuse
- cos(θ) = adjacent/hypotenuse
- tan(θ) = opposite/adjacent
- Pythagorean identity: sin²(θ) + cos²(θ) = 1.
- Inverse trig: θ = sin⁻¹(x) finds the angle.
In a right triangle, if the opposite side = 3 and hypotenuse = 5: sin(θ) = 3/5 = 0.6.
Practice — Lessons 1–3
Q1 · Multiple Choice
In a right triangle, cos(θ) = 5/13. What is tan(θ)?
Q2 · Short Answer
If sin(θ) = 0.6 and cos(θ) = 0.8, what is tan(θ)?
04–05 — Lessons 4–5
Radian Measure
Radians are an alternative unit for measuring angles based on arc length.
- 1 radian: angle where arc length = radius.
- π radians = 180°
- To convert: degrees × π/180 = radians.
- To convert: radians × 180/π = degrees.
- Full circle: 2π radians = 360°.
90° = π/2 · 45° = π/4 · 60° = π/3 · 30° = π/6
Practice — Lessons 4–5
Q3 · Multiple Choice
Convert 270° to radians.
Q4 · Short Answer
Convert π/3 radians to degrees.
Topic B: The Unit Circle
Lessons 6–9 · Unit circle coordinates, reference angles, trig values.
06–09 — Lessons 6–9
The Unit Circle
The unit circle has radius 1. Every point (x, y) on it gives cos(θ) = x and sin(θ) = y.
- Key angles (radians): 0, π/6, π/4, π/3, π/2, π, 3π/2, 2π.
- Quadrant I: sin > 0, cos > 0.
- Quadrant II: sin > 0, cos < 0.
- Quadrant III: sin < 0, cos < 0.
- Quadrant IV: sin < 0, cos > 0.
- ASTC: All Students Take Calculus — which trig values are positive.
cos(π/3) = 1/2 · sin(π/3) = √3/2 · tan(π/3) = √3
Practice — Lessons 6–9
Q5 · Multiple Choice
What is sin(π/2)?
Q6 · Multiple Choice
In which quadrant is θ if sin(θ) < 0 and cos(θ) > 0?
Topic C: Graphing Trigonometric Functions
Lessons 10–14 · Graphs of sin/cos/tan, amplitude, period, and transformations.
10–12 — Lessons 10–12
Graphs of Sine & Cosine
- f(x) = A·sin(Bx + C) + D
- Amplitude = |A|: half the vertical distance from min to max.
- Period = 2π/B: horizontal length of one full cycle.
- Phase shift = −C/B: horizontal translation.
- Vertical shift = D: moves midline up/down.
- Cosine starts at max; Sine starts at midline.
f(x) = 3sin(2x) → A=3 (amplitude 3), B=2 (period = 2π/2 = π).
Practice — Lessons 10–12
Q7 · Multiple Choice
For f(x) = 4cos(πx/2), what is the period?
Q8 · Short Answer
For f(x) = 2sin(3x), what is the amplitude?
13–14 — Lessons 13–14
The Tangent Function
The tangent function has a period of π and vertical asymptotes where cos(θ) = 0.
- tan(θ) = sin(θ)/cos(θ)
- Period: π.
- Vertical asymptotes: x = π/2 + nπ for integers n.
- Increases from −∞ to ∞ on each period.
- No amplitude — unbounded.
tan(0) = 0 · tan(π/4) = 1 · tan(π/2) is undefined.
Practice — Lessons 13–14
Q9 · Multiple Choice
Where does f(x) = tan(x) have a vertical asymptote?
Q10 · Multiple Choice
What is the period of f(x) = tan(x)?