How to Find Domain and Range of Trigonometric Functions?

The domain and range of trigonometric functions can be found by examining where the function is defined and the function’s output values for each input value.

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A step-by-step guide to the domain and range of trigonometric functions

The domain and range of trigonometric functions are the input values and the output values of trigonometric functions, respectively. The domain of trigonometric functions shows the values of the angles in which the trigonometric functions are defined. The range of trigonometric functions shows the value of the result of the trigonometric function corresponding to a specific angle in the domain.

 Sine function:

We know that the sine function is the ratio of the perpendicular and hypotenuse of a right-angled triangle. The domain and range of trigonometric function sine are obtained as follows:

• Domain \(=\) \(\color{blue}{All\:real\:numbers,\:i.e.,\:(−∞,\:∞)}\)
• Range \(=\color{blue}{ [-1, 1]}\)

Cosine function:

The cosine function is the ratio of the adjacent side and the hypotenuse of a right triangle. The domain and range of trigonometric function cosine are obtained as follows:

• Domain \(=\) \(\color{blue}{All\:real\:numbers,\:i.e.,\:(−∞,\:∞)}\)
• Range \(=\color{blue}{ [-1, 1]}\)

Tangent function:

The tangent function can be written as the ratio of the sine and cosine functions, so the domain of \(tan x\) does not contain values where \(cos x\) is zero. We know that \(cos x\) is \(0\) in odd integral multiples of \(\frac{π}{2}\), so the domain and range of trigonometric function tangent are obtained as follows:

• Domain \(=\) \(\color{blue}{R – (2n + 1)\frac{π}{2}}\)
• Range \(=\) \(\color{blue}{(−∞, ∞)}\)

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Cotangent function:

The cotangent function can be written as the ratio of the cosine and the sine function, and \(cot x\) is the reciprocal of \(tan x\). Therefore, the range of \(cot x\) does not contain values where \(sin x\) is equal to zero. We know that \(sin x\) in integral multiples of \(π\) is \(0\), so the domain and range of trigonometric function cotangent are given by:

• Domain \(=\)\(\color{blue}{R – nπ}\)
• Range \(=\) \(\color{blue}{(−∞, ∞)}\)

Secant function:

It can also be written as the reciprocal of the cosine function. Therefore, the domain of sec x does not contain values where \(cos x\) is equal to zero. \(cos x\) is \(0\) in odd integral multiples of \(π\), so the domain and range of trigonometric function secant are obtained as follows:

• Domain \(=\) \(\color{blue}{R – (2n + 1)\frac{π}{2}}\)
• Range \(=\) \(\color{blue}{(-∞, -1] \cup [+1, +∞)}\)

Cosecant function:

The cosecant function can also be written as the reciprocal of the sine function. Therefore the domain of trigonometric function \(cosec x\) does not contain values where \(sin x\) is equal to zero. We know that \(sin x\) is \(0\) at integral multiples of \(π\), hence the domain and range of trigonometric function cosecant are given by:

• Domain \(=\) \(\color{blue}{R – nπ}\)
• Range \(=\) \(\color{blue}{(-∞,\:-1]\:\cup \:[+1,\:+∞)}\)

Domain and Range of Trigonometric Functions, Example 1:

Find the domain and range of \(y= cos(x)\: – 3\)

Solution:

• Domain: \(\:\left(-\infty \:,\:\infty \:\right)\)
• Range: \(-1\le \:cos\:x\le \:1\:⇒-1\:-\:3\le \:cos\:x-3\le \:1-3\:⇒\:-4\le \:y\le -2\:\)

Domain and Range of All Six Trig Functions (Table)

Here is every result on one screen. In the table, n stands for any integer.

A quick way to remember the gaps: tan and sec divide by cos x, so they break where cos x = 0. Cot and csc divide by sin x, so they break where sin x = 0.

Domain and Range of Inverse Trig Functions

A trig function repeats its outputs, so it has no inverse on its full domain. We fix this by keeping one piece of each graph. That chosen piece sets the range of the inverse, often called the principal values.

Example: Find the domain of y = sin⁻¹(2x). The input to sin⁻¹ must stay between −1 and 1, so −1 ≤ 2x ≤ 1. Divide by 2: −½ ≤ x ≤ ½. The range is still [−π/2, π/2].

Example: Find the range of y = 3 sin x + 1. Start with −1 ≤ sin x ≤ 1. Multiply by 3: −3 ≤ 3 sin x ≤ 3. Add 1: −2 ≤ y ≤ 4. The range is [−2, 4], and the domain is all real numbers.

Frequently Asked Questions

How do you find the domain and range of sine and cosine functions?

Sine and cosine accept any real number as input, so their domain is all real numbers. Their outputs are the y- and x-coordinates of points on the unit circle, so the range is [−1, 1]. For example, sin 90° = 1 and cos 180° = −1 are the largest and smallest values they can reach.

How do you find the domain and range of tangent functions?

Tangent is sin x/cos x, so it is undefined where cos x = 0. Its domain is all real numbers except x = π/2 + nπ, where n is any integer, and its graph has vertical asymptotes there. Between the asymptotes it takes every value, so the range of y = tan x is all real numbers.

How do you find the domain and range of inverse trig functions?

Swap the domain and range of the restricted original function. Sine restricted to [−π/2, π/2] has range [−1, 1], so arcsin x has domain [−1, 1] and range [−π/2, π/2]. Arccos x has domain [−1, 1] and range [0, π], and arctan x has domain all real numbers and range (−π/2, π/2).

How do you find the domain and range of a trig graph?

Read the domain left to right and the range bottom to top, or start from the basic range and apply each transformation. For y = 3 sin x + 2, start with −1 ≤ sin x ≤ 1, multiply by 3 to get −3 ≤ 3 sin x ≤ 3, then add 2. The range is [−1, 5], and the domain is still all real numbers.

What are the domain and range of secant, cosecant and cotangent?

Secant, 1/cos x, excludes x = π/2 + nπ and has range (−∞, −1] ∪ [1, ∞). Cosecant, 1/sin x, excludes x = nπ and has the same range as secant. Cotangent, cos x/sin x, also excludes x = nπ, but its range is all real numbers.