The Tangent Function | Moosa Academy

Tangent is not a wave between −1 and 1 like sine and cosine. It is their quotient, and that single fact explains everything about it: the vertical asymptotes, the shorter period, and the unlimited range.

Concept Definition

\tan\theta = \frac{\sin\theta}{\cos\theta}

In a right triangle this is the same as $\dfrac{\text{opposite}}{\text{adjacent}}$ .

Wherever $\cos\theta = 0$ — that is at 90°, 270°, and generally $90° + 180°n$ — the quotient divides by zero, so $\tan\theta$ is undefined. Those angles become vertical asymptotes on the graph.

Graph Branches and asymptotes

Each branch climbs from far below to far above, then the pattern restarts. The distance from one branch to the next is only 180°, so the period of tangent is 180° = π — half the period of sine and cosine. The range is all real numbers.

Note Sign by quadrant

Positive in the first and third quadrants: there sine and cosine share the same sign, so their quotient is positive.

Negative in the second and fourth quadrants, where sine and cosine have opposite signs. Tangent is also an odd function: $\tan(-\theta) = -\tan\theta$ , symmetric about the origin.

Example Three worked cases

Find

\tan 240°

. Since

240° = 180° + 60°

, the angle is in the third quadrant, where tangent is positive, and the reference angle is 60°.

⟹ tan 240° = √3 ≈ 1.732

Find

\tan 135°

. Second quadrant, so tangent is negative; the reference angle is

180° - 135° = 45°

⟹ tan 135° = −1

Simplify

\tan\theta \times \cos\theta = \dfrac{\sin\theta}{\cos\theta} \times \cos\theta

⟹ = sin θ

Reference Properties of tan θ

Definition	tan θ = sin θ ÷ cos θ
Domain	all angles except 90° + 180°n
Range	all real numbers
Period	180° = π
Symmetry	odd: tan(−θ) = −tan θ
Asymptotes	wherever cos θ = 0

Summary

tan θ = sin θ ÷ cos θ, or opposite ÷ adjacent in a right triangle.
Its period is only 180° — half that of sine and cosine.
Vertical asymptotes appear at 90° + 180°n, where cos θ = 0.
It is positive in quadrants I and III, negative in II and IV, and odd about the origin.