AP PRECALCULUS — UNIT 3A · TRIGONOMETRIC & POLAR FUNCTIONS
3.2B — Sine, Cosine, and Tangent
Notes — The Three Primary Trig Functions
💡 Learning Objectives (3.2.A Part 2)
By the end of this lesson you will be able to:
- Define sine, cosine, and tangent in terms of a point on the unit circle
- Relate the unit-circle definitions to the right-triangle definitions
- Use sign patterns across the four quadrants to determine the sign of each trig value
- Evaluate sin(θ), cos(θ), and tan(θ) for common angles
1. The Unit-Circle Definition
Place an angle θ in standard position. Let (x, y) be the point where its terminal ray meets the UNIT CIRCLE (the circle of radius 1). Then:
cos(θ) = x, sin(θ) = y, tan(θ) = y/x (when x ≠ 0)
This is the fundamental definition. The cosine of an angle is the x-coordinate of the point on the unit circle; the sine is the y-coordinate. Tangent is their ratio.
2. Connection to Right Triangles
If θ is an acute angle in a right triangle, the three trig ratios can also be defined as:
- sin(θ) = opposite / hypotenuse
- cos(θ) = adjacent / hypotenuse
- tan(θ) = opposite / adjacent
When you drop a perpendicular from the unit-circle point (x, y) to the x-axis, you create a right triangle with hypotenuse 1, opposite leg y, adjacent leg x. The two definitions agree.
📝 Why extend beyond triangles
Right-triangle definitions only make sense for acute angles (0 to π/2). The unit-circle definition works for ALL real numbers, positive, negative, or huge — and that is what makes sine, cosine, and tangent into functions of any real input.
3. Signs by Quadrant
Since cos(θ) = x and sin(θ) = y on the unit circle, the signs of these values match the signs of the coordinates in each quadrant:
Quadrant | sin | cos | tan |
|---|---|---|---|
I (0 to π/2) | + | + | + |
II (π/2 to π) | + | − | − |
III (π to 3π/2) | − | − | + |
IV (3π/2 to 2π) | − | + | − |
Memory trick: ‘All Students Take Calculus’ — in Quadrant I All are positive, in II Sine, in III Tangent, in IV Cosine.
4. Values at the Quadrantal Angles
At 0, π/2, π, and 3π/2 the terminal ray lies along an axis, so (x, y) is one of (1,0), (0,1), (−1,0), (0,−1).
θ | 0 | π/2 | π | 3π/2 | 2π |
|---|---|---|---|---|---|
cos θ | 1 | 0 | −1 | 0 | 1 |
sin θ | 0 | 1 | 0 | −1 | 0 |
tan θ | 0 | undef | 0 | undef | 0 |
⚠️ Why tangent is undefined at π/2
tan(θ) = y/x, and at π/2 we have x = 0. Division by zero is undefined — there is no tangent there. The same happens at 3π/2 and at every θ of the form π/2 + kπ.
5. Special Acute Angles
The three ‘special’ acute angles produce exact, memorizable values. These come from 30-60-90 and 45-45-90 right triangles.
θ | π/6 (30°) | π/4 (45°) | π/3 (60°) |
|---|---|---|---|
cos θ | √3/2 | √2/2 | 1/2 |
sin θ | 1/2 | √2/2 | √3/2 |
tan θ | √3/3 | 1 | √3 |
Tip: the sines go 1/2, √2/2, √3/2 — think of them as √1/2, √2/2, √3/2.
6. Evaluating in Other Quadrants
For any angle θ, use this procedure:
- Find the reference angle θ_ref (the acute angle from the x-axis)
- Look up sin(θ_ref), cos(θ_ref), tan(θ_ref) — these are always positive
- Use the quadrant sign chart to attach the correct sign
📘 Example — cos(5π/6) and sin(11π/6)
- 5π/6: Quadrant II (cos negative). Reference angle = π/6. cos(π/6) = √3/2. So cos(5π/6) = −√3/2
- 11π/6: Quadrant IV (sin negative). Reference angle = π/6. sin(π/6) = 1/2. So sin(11π/6) = −1/2
7. The Pythagorean Identity
Because every point (cos θ, sin θ) lies on the unit circle x² + y² = 1, we get the most important trig identity:
cos²(θ) + sin²(θ) = 1
This holds for every real θ. It lets you find one of the two values from the other (up to a sign determined by the quadrant).
8. Summary
- On the unit circle: cos(θ) = x-coordinate, sin(θ) = y-coordinate, tan(θ) = y/x
- Right-triangle definitions agree with unit-circle ones for acute θ
- Signs follow the quadrant: ‘All Students Take Calculus’
- Memorize the three special-angle rows and the four quadrantal columns
- cos²(θ) + sin²(θ) = 1 for every θ — the Pythagorean identity