AP PRECALCULUS — UNIT 3B · TRIGONOMETRIC & POLAR FUNCTIONS
3.8 — The Tangent Function
Notes — Slopes Around the Unit Circle
💡 Learning Objectives (3.8.A)
By the end of this lesson you will be able to:
- Define tan(θ) using the unit-circle ratio sin(θ)/cos(θ)
- Identify the period, vertical asymptotes, and zeros of y = tan(x)
- Sketch y = tan(x) over multiple periods and identify increasing intervals
- Apply transformations to the tangent function
1. Tangent as a Ratio
On the unit circle, tan(θ) = sin(θ) / cos(θ) = y/x. Geometrically, tan(θ) is the SLOPE of the line from the origin to the point (cos θ, sin θ). When θ = 0 the slope is 0 (the line is horizontal). When θ = π/4 the slope is 1. When θ approaches π/2, the line becomes vertical and tan(θ) blows up to infinity.
tan(θ) = sin(θ) / cos(θ) for every θ where cos(θ) ≠ 0
2. Where Is Tangent Undefined?
Because tan(θ) = sin(θ)/cos(θ), tangent is undefined wherever cos(θ) = 0. These are the values θ = π/2, 3π/2, 5π/2, … and their negatives. In short:
tan(θ) is undefined at θ = π/2 + kπ for every integer k.
These x-values become VERTICAL ASYMPTOTES on the graph of y = tan(x). The function shoots up to +∞ as x approaches π/2 from the left, and shoots down from −∞ as x leaves π/2 going right.
3. Period of Tangent
Surprisingly, tangent has a SHORTER period than sine and cosine. Both numerator (sin) and denominator (cos) flip sign together when you add π, so their ratio stays the same:
- sin(θ + π) = −sin(θ)
- cos(θ + π) = −cos(θ)
- tan(θ + π) = (−sin θ)/(−cos θ) = tan(θ)
The period of y = tan(x) is π — half the period of sine or cosine.
4. Key Values
θ | 0 | π/6 | π/4 | π/3 | π/2 | 2π/3 | 3π/4 | 5π/6 | π |
|---|---|---|---|---|---|---|---|---|---|
tan θ | 0 | √3/3 | 1 | √3 | undef | −√3 | −1 | −√3/3 | 0 |
Tangent goes from 0 up to +∞ on (0, π/2), then jumps to −∞ and rises through 0 again on (π/2, π). The pattern repeats every π units.
5. Sketching y = tan(x)
Over the interval (−π/2, π/2) — one ‘standard’ period — the graph:
- has vertical asymptotes at x = −π/2 and x = π/2
- passes through (0, 0)
- passes through (−π/4, −1) and (π/4, 1)
- is INCREASING throughout the interval
- has range of all real numbers
The graph repeats the same shape on every interval of length π between consecutive asymptotes.
6. Symmetry and End Behavior
- Tangent is ODD: tan(−θ) = −tan(θ). The graph has origin symmetry.
- As x → (π/2)⁻, tan(x) → +∞
- As x → (π/2)⁺, tan(x) → −∞
- Range: all real numbers
- Domain: all reals except π/2 + kπ
7. Transformations of Tangent
The general form is y = a · tan(b(x − c)) + d, with the same parameter meanings as for sine:
- a — vertical scale factor (amplitude is not really meaningful for tangent since the range is unbounded)
- b — horizontal scale factor; period becomes π/|b|
- c — phase shift (right by c)
- d — vertical shift
⚠️ Common mistake
Tangent's period is π, not 2π. So for y = tan(bx) the period is π/|b|, not 2π/|b|. Don't apply the sinusoidal formula here.
8. Worked Example
📘 Example — Sketch y = 2 tan((π/4)(x − 1))
- Period: π / (π/4) = 4. Asymptotes are 4 units apart.
- Phase shift: c = 1. Center of one period is at x = 1; asymptotes at x = 1 − 2 = −1 and x = 1 + 2 = 3.
- Vertical scale: a = 2. The point that was (π/4, 1) on the parent becomes (1 + 1, 2) = (2, 2).
- Sketch goes up steeply between the asymptotes, passing through (1, 0) and (2, 2).
9. Summary
- tan(θ) = sin(θ)/cos(θ); slope of the radius line on the unit circle
- Period π; vertical asymptotes at π/2 + kπ; zeros at kπ
- Increasing on every open interval between consecutive asymptotes; range is all reals
- Tangent is an odd function with origin symmetry
- Use y = a · tan(b(x − c)) + d for transformations; period is π/|b|