The Tangent Function

tan(x) = sin(x) / cos(x). That's the whole definition. When cos(x) = 0, you're dividing by zero — and that's exactly where the asymptotes live. Period π, not 2π. Unbounded range. No amplitude. The most different of the three.

tan(x) = sin(x) / cos(x) — tangent is the ratio of the y-coordinate to the x-coordinate on the unit circle. It's rise over run. When cos = 0 (at π/2 and 3π/2), the denominator vanishes and tan blows up to ±∞.

Definition

sin(x)/cos(x)

Period

π (not 2π!)

Domain

x ≠ π/2 + nπ

Range

(−∞, +∞)

Amplitude

None (unbounded)

Odd function

tan(−x) = −tan(x)

⚠️

Common mistakeThe period of tan is π, not 2π. Sin and cos need a full lap around the unit circle to repeat. Tan repeats every half lap — from one asymptote to the next, a distance of π.

Why tan blows up — asymptote derivation

tan(x) = sin(x) / cos(x). Division by zero is undefined.

cos(x) = 0 at x = π/2 and x = 3π/2 (and every π after that). Formula: x = π/2 + nπ, n any integer.

As cos(x) → 0⁺, sin(x) stays near ±1 — so the fraction approaches ±∞. That vertical blow-up is the asymptote.

The graph never touches these vertical lines. It approaches them and shoots off to infinity on both sides.

🔬

Lab crossoverThink of tan like a pH reading at exactly 7.00 with a broken probe — you know it's the inflection point, but the reading goes haywire right at the critical value. Tan is undefined the same way: the function approaches that vertical wall and the output explodes.

Key values table

x (deg)	x (rad)	sin(x)	cos(x)	tan(x)
0°	0	0	1	0
30°	π/6	1/2	√3/2	√3/3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	undefined
120°	2π/3	√3/2	−1/2	−√3
135°	3π/4	√2/2	−√2/2	−1
180°	π	0	−1	0

Memory anchors

tan = 1 at π/4

At 45°, sin and cos are equal (both √2/2), so their ratio is always 1. The "balanced" point.

tan = 0 at every nπ

tan = 0 whenever sin = 0, at 0, π, 2π, −π... These are the x-intercepts.

Period is π, not 2π

One full cycle: −π/2 to π/2. From one asymptote to the next is exactly π.

Odd function

tan(−x) = −tan(x). The graph has 180° rotational symmetry about the origin.

Adjust periods and zoom to see how the graph behaves. Watch what happens near the asymptotes as you zoom in.

Periods shown: 2 y-limit: 6

y = tan(x) Asymptote Zero crossings

Period

π ≈ 3.14159

Zero crossings

x = nπ

Asymptotes

x = π/2 + nπ

tan = 1 at

x = π/4 + nπ

Transformations: y = A·tan(Bx − C) + D

Parameter	What it does	Example
A	Vertical stretch. Negative A flips the graph.	y = 2tan(x) → twice as steep
B	New period = π/\|B\|	y = tan(2x) → period = π/2
C	Phase shift = C/B	y = tan(x − π/4) → shifted right π/4
D	Vertical shift of midline	y = tan(x) + 1 → midline at y = 1

📌

New asymptote locations with By = tan(Bx) → asymptotes at x = π/(2B) + nπ/B. The period and asymptote spacing compress by the same factor as B.

Tan vs sin vs cos — three trig functions, three very different shapes and properties.

tan(x)

Period: π
Range: (−∞, ∞)
No amplitude
Asymptotes at π/2 + nπ
Odd: tan(−x) = −tan(x)
Zeros at nπ

sin(x)

Period: 2π
Range: [−1, 1]
Amplitude: 1
No asymptotes
Odd: sin(−x) = −sin(x)
Zeros at nπ

cos(x)

Period: 2π
Range: [−1, 1]
Amplitude: 1
No asymptotes
Even: cos(−x) = cos(x)
Zeros at π/2 + nπ

💡

Key overlapSin and tan both have zeros at nπ. But cos has zeros exactly where tan has asymptotes — x = π/2 + nπ. Makes sense: tan = sin/cos, so tan blows up the moment cos hits zero.

🔄

Tan and cot — the reciprocal pair

▼

cot(x) = 1/tan(x) = cos(x)/sin(x). Where tan has zeros (at nπ), cot has asymptotes. Where tan has asymptotes (at π/2 + nπ), cot has zeros. The graph of cot is essentially tan flipped upside down and shifted.

Property	tan(x)	cot(x)
Period	π	π
Asymptotes	x = π/2 + nπ	x = nπ
Zeros	x = nπ	x = π/2 + nπ
Slope direction	Increasing	Decreasing

🔢

arctan — the restricted inverse

▼

To make an inverse, tan must be restricted to where it's one-to-one: (−π/2, π/2) — one full period, not including the asymptotes. arctan takes any real number and returns an angle in that range.

Domain of arctan

(−∞, ∞)

Range of arctan

(−π/2, π/2)

arctan(0)

arctan(1)

π/4

arctan(√3)

π/3

arctan(−1)

−π/4

Five practice problems. Try each one before revealing the answer.

Evaluate tan(π/3). Show your reasoning using sin and cos.

tan(π/3) = √3
sin(π/3) = √3/2 and cos(π/3) = 1/2.
tan(π/3) = sin/cos = (√3/2) ÷ (1/2) = √3 ≈ 1.732.

Where are the vertical asymptotes of y = tan(x) in the interval [−2π, 2π]?

x = −3π/2, −π/2, π/2, 3π/2
Formula: x = π/2 + nπ. Plugging in n = −2, −1, 0, 1 gives those four values. Always use the formula rather than memorizing each one.

For y = tan(2x), what is the period and where is the first positive asymptote?

Period = π/2. First positive asymptote at x = π/4.
Period formula: π/|B| = π/2.
Asymptote: x = π/(2B) + nπ/B = π/4 + nπ/2. For n = 0: x = π/4.

Is tan(x) even, odd, or neither? How can you tell from the graph?

Odd. tan(−x) = −tan(x).
Algebraically: tan(−x) = sin(−x)/cos(−x) = −sin(x)/cos(x) = −tan(x).
Graphically: the curve has 180° rotational symmetry about the origin. Spin it halfway around and it lands on itself.

What is arctan(−√3)? Give your answer in radians in the restricted range.

−π/3
We need the angle in (−π/2, π/2) whose tangent is −√3. Since tan(π/3) = √3 and tan is odd, tan(−π/3) = −√3. The restricted range includes −π/3.

Quick reference

tan = sin/cos
Period = π
Asymptotes: x = π/2 + nπ
Zeros: x = nπ
y = A·tan(Bx): period = π/|B|

Special values:
tan(0) = 0
tan(π/6) = √3/3
tan(π/4) = 1
tan(π/3) = √3
tan(π/2) = undefined

Key vocabulary

Vertical asymptote

A vertical line the graph approaches but never touches. tan has asymptotes at x = π/2 + nπ (where cos = 0).

Period

Horizontal length of one complete cycle. tan: π. sin and cos: 2π. The B parameter changes period to π/|B|.

Odd function

f(−x) = −f(x). Symmetric about the origin. tan, sin are odd. cos is even.

Amplitude

Half the distance between max and min. tan has NO amplitude — it's unbounded. Only sin and cos have amplitude.

arctan / tan⁻¹

Inverse of tan. Domain: all reals. Range: (−π/2, π/2). Always outputs an angle in QI or QIV.

Exam reminders

Period of tan = π, not 2π. This is the most common mistake.

Asymptotes at x = π/2 + nπ — where cos = 0.

tan has no amplitude — don't write one. Undefined/none.

y = tan(Bx) → period = π/|B|, asymptotes at π/(2B) + nπ/B.

tan(−x) = −tan(x) → odd function → 180° rotational symmetry.

arctan range: (−π/2, π/2) open — never reaches ±π/2.