Mathfolis

The Tangent Function

Unit 3 · Trigonometric and Polar Functions

What AP Precalc asks here

The tangent function is tan⁡x=sin⁡x/cos⁡x\tan x = \sin x / \cos x. It has period π\pi (half that of sine and cosine), vertical asymptotes wherever cos⁡x=0\cos x = 0 (i.e., x=π/2+nπx = \pi/2 + n\pi), and zeros wherever sin⁡x=0\sin x = 0 (i.e., x=nπx = n\pi). 'Amplitude' is meaningless for tan⁡\tan — its range is all real numbers.

Properties of y=tan⁡xy = \tan x

Definition
tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x}
Period
π\pi
Vertical asymptotes
x=π2+nπx = \tfrac{\pi}{2} + n\pi
Zeros
x=nπx = n\pi

Special values

tan⁡0=0\tan 0 = 0
tan⁡(π/6)=3/3\tan(\pi/6) = \sqrt{3}/3
tan⁡(π/4)=1\tan(\pi/4) = 1
tan⁡(π/3)=3\tan(\pi/3) = \sqrt{3}
tan⁡(π/2)\tan(\pi/2) undefined
Caution: tan does not have an amplitude — its range is all real numbers. The 'A' in Atan⁡(B(x−h))+kA \tan(B(x - h)) + k is a vertical stretch, not an amplitude bound.
Type 1

Graph and key features

Period is π\pi. Asymptotes happen where cos⁡x=0\cos x = 0.

Example 1
What is the period of y=tan⁡xy = \tan x? (A) π/2\pi/2 (B) π\pi (C) 2π2\pi (D) 4π4\pi

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Type 2

Exact values at special angles

Use tan⁡=sin⁡/cos⁡\tan = \sin / \cos at the special angles.

Example 2
What is tan⁡(π/3)\tan(\pi/3)? (A) 1/31/\sqrt 3 (B) 11 (C) 3\sqrt 3 (D) 3/2\sqrt 3 / 2

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Type 3

Transformations of tangent

y=Atan⁡(Bx)y = A \tan(B x) has period π/∣B∣\pi / |B|.

Example 3
What is the period of y=tan⁡(2x)y = \tan(2x)? (A) π/4\pi/4 (B) π/2\pi/2 (C) π\pi (D) 2π2\pi

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