# Domain and Range of Trigonometric Functions

TL;DR

The domain and range of trigonometric functions describes which angles each function accepts and which output values it produces — sine and cosine accept all real angles and output values in \[-1,1\], while tangent, cotangent, secant, and cosecant have angles where they are undefined. This article gives the full domain–range table, the graph of each function in degrees and radians, the unit-circle anchor for each definition, three worked examples, and the most common mistakes students make.

## Six Functions, Three Different Output Patterns, One Unit Circle

Every trigonometric function inherits its domain and range directly from the unit circle — and those six rules close together in a single table.

The **domain** of a function is the set of inputs it accepts; the **range** is the set of outputs it returns. For trigonometric functions, the input is an angle (in degrees or radians) and the output is a real number tied to the unit circle.

## The Master Table

For all six trigonometric functions, the domain and range are below. Real-number set is \(R\); integer set is \(Z\).

| Function                 | Domain (in radians)                      | Domain (in degrees)        | Range                   |
|--------------------------|-------------------------------------------|----------------------------|-------------------------|
| sin(θ)                   | θ ∈ R                                    | all real angles            | \[-1,1\]                |
| cos(θ)                   | θ ∈ R                                    | all real angles            | \[-1,1\]                |
| tan(θ)                   | θ ≠ (2n+1)π/2, n ∈ Z                    | θ ≠ 90°, 270°, …          | R                       |
| cot(θ)                   | θ ≠ nπ, n ∈ Z                           | θ ≠ 0°, 180°, …           | R                       |
| sec(θ)                   | θ ≠ (2n+1)π/2, n ∈ Z                    | θ ≠ 90°, 270°, …          | (−∞,−1] ∪ [1,∞)       |
| csc(θ)                   | θ ≠ nπ, n ∈ Z                           | θ ≠ 0°, 180°, …           | (−∞,−1] ∪ [1,∞)       |

Three patterns sit inside this table:

- **Sine and cosine** — always defined, always bounded in \[-1,1\].
- **Tangent and cotangent** — undefined at the angles where the relevant unit-circle coordinate is zero; output sweeps all of R.
- **Secant and cosecant** — undefined at the same angles as their reciprocal cousins; output never lies inside (−1,1), since reciprocals of values ≤1 in magnitude land ≥1 in magnitude.

> **Quick facts.**  
> - **Period:** sin(θ), cos(θ), sec(θ), csc(θ) have period 2π (360°). tan(θ) and cot(θ) have period π (180°).  
> - **Undefined points come from zero denominators.** tan(θ) = sin(θ)/cos(θ) is undefined when cos(θ) = 0; sec(θ) = 1/cos(θ) is undefined at the same angles.

## Double-Anchoring — Right Triangle and Unit Circle

For an angle in (0,π/2), every function has both a right-triangle reading and a unit-circle reading. Domain and range come cleanly from the unit-circle view.

**Sine and cosine.** On the unit circle, a point at angle θ has coordinates (cos(θ), sin(θ)). The coordinates of any point on the unit circle are bounded by ±1, so the range is \[-1,1\]. The angle θ can be any real number, so the domain is R.

**Tangent.** tan(θ) = sin(θ)/cos(θ) = y-coord/x-coord. This is undefined when the x-coordinate equals zero — at θ=π/2,3π/2,… (90°,270°,…). Where defined, the ratio sweeps all real numbers — vertical asymptotes appear at the excluded angles.

**Secant.** sec(θ) = 1/cos(θ) = 1/(x-coord). Undefined where the x-coordinate is zero. When defined, |cos(θ)| ≤ 1 ⇒ |sec(θ)| ≥ 1.

**Cotangent.** cot(θ) = cos(θ)/sin(θ) = x/y — undefined where the y-coordinate is zero, at θ=0,π,2π,…

**Cosecant.** csc(θ) = 1/sin(θ) = 1/y — undefined where y=0, range matches secant.

The unit circle is the **one figure** that resolves all six domain–range rules in one go.

## Function-by-Function with Graphs

### Sine — sin(θ)

- **Domain:** θ ∈ R (all real numbers).
- **Range:** \[-1,1\].
- **Period:** 2π or 360°.
- **Reference points:** sin(0) = 0, sin(π/2) = 1, sin(π) = 0, sin(3π/2) = −1.

### Cosine — cos(θ)

- **Domain:** θ ∈ R.
- **Range:** \[-1,1\].
- **Period:** 2π or 360°.
- **Reference points:** cos(0) = 1, cos(π/2) = 0, cos(π) = −1, cos(3π/2) = 0.

Cosine has the same shape as sine, shifted left by π/2: cos(θ) = sin(θ + π/2).

### Tangent — tan(θ)

- **Domain:** θ ∈ R ∖ {(2n+1)π/2:n ∈ Z}. In degrees, θ ≠ 90°, 270°, 450°, …
- **Range:** R (all real numbers).
- **Period:** π or 180°.

### Cotangent — cot(θ)

- **Domain:** θ ≠ nπ, n ∈ Z. In degrees, θ ≠ 0°, 180°, 360°, …
- **Range:** R.
- **Period:** π.

### Secant — sec(θ) = 1/cos(θ)

- **Domain:** same as tangent — θ ≠ (2n+1)π/2.
- **Range:** (−∞,−1] ∪ [1,∞) — never lies in (−1,1).
- **Period:** 2π.

### Cosecant — csc(θ) = 1/sin(θ)

- **Domain:** same as cotangent — θ ≠ nπ.
- **Range:** (−∞,−1] ∪ [1,∞).
- **Period:** 2π.

## Three Worked Examples — Quick, Standard, Stretch

**Quick.** Find the domain and range of f(θ) = 2sin(θ).

The domain of sin(θ) is R, and multiplying by 2 doesn't restrict inputs, so the domain remains R. The range is \[-1,1\], so multiplying by 2 stretches the range to \[-2,2\].

**Final answer:** Domain R; Range \[-2,2\].

**Standard (Wrong Path First — Where Intuition Breaks).** Find the domain of g(θ) = tan(2θ−π/3).

_The wrong path._ A student writes "the domain of tan is R∖{(2n+1)π/2}, so the domain of tan(2θ−π/3) is also R∖{(2n+1)π/2}."

**The correct way:** Solve the input transformation: 2θ − π/3 = (2n+1)π/2, not just copied from the bare function's exclusions.

**Final answer:** Domain is θ ∈ R ∖ {f(θ) solution}; Range is R.

**Stretch.** Find the domain and range of h(θ) = 3/(2 + cos(θ)).

**Domain:** The denominator 2 + cos(θ) must be nonzero. Since cos(θ) ∈ \[-1,1\], the denominator 2 + cos(θ) ∈ [1,3] — always positive, never zero.
**Range:** As cos(θ) sweeps \[-1,1\], the function h(θ) sweeps \[1,3\].

**Final answer:** Domain R. Range \[1,3\].

## Why Domain and Range Matter Outside the Classroom

Domain and range aren't just textbook bookkeeping — they cause real outages when missed.

- **Signal processing.** Audio codecs treat amplitude as sin/cos outputs in \[-1,1\] — sending a value ±1.2 into a fixed-point DAC produces clipping.
- **Computer graphics.** Direction vectors normalized onto a unit sphere use θ=arccos(z); z outside \[-1,1\] crashes the renderer. Wrap every arccos in a `clamp(z, -1, 1)`.
- **GPS satellite ranging.** Computing the angle between a satellite vector and a receiver vector uses arccos — must protect the input from drifting outside \[-1,1\].
- **Phasor analysis in electrical engineering.** Sinusoidal AC currents are modeled as I(t)=I0sin(ωt + ϕ); peak current multiplies the trig range \[-1,1\] to give \[-I0,I0\].
- **Astronomy.** The hour angle of a star has a 24-hour periodicity, using tangent's period π — observatory software respects tangent's excluded angles.

## The Mathematicians Who Mapped the Trig Functions

**Aryabhata (476–550 CE, India)** introduced the half-chord function _jya_ (modern sine) and tabulated it across what we'd now call the first quadrant.  
**Bhaskara II (1114–1185, India)** extended trig tables across the full circle.

> **Leonhard Euler (1707–1783, Switzerland).** Euler was the first to treat trigonometric functions as functions of a real variable.

## Slip-Ups That Cost Marks

### **1. Forgetting tangent's undefined points.**
- Assume all six trig functions have the same domain.

### **2. Confusing the range of secant with the range of cosine.**
- Don't reach for the cosine range when the function is its reciprocal.

### **3. Treating composite trig functions as if they kept the base domain.**
- Always solve for the actual excluded angles.

### **4. Mode confusion: writing the domain in degrees but solving in radians (or vice versa).** 
- Pick a unit and stick with it.

## Conclusion

The **domain and range of trigonometric functions** follow three patterns: sin/cos are defined everywhere with range \[-1,1\]; tan/cot have range R but are undefined at half-period intervals; sec/csc have range outside (−1,1) and inherit the tan/cot exclusions.