Question

Where is the tangent function undefined?

Short answer

Answer: The tangent function is undefined at $$x=\frac{(2n+1)\pi}{2}$$ where n is an integer because the cosine function is equal to zero at these values and division by zero is undefined.

Step-by-step solution

Identify the denominator of the tangent function

The tangent function is given by tan(x) = sin(x)/cos(x). The denominator is the cosine function, cos(x).

Find values of x where cos(x) = 0

Cosine function, cos(x), is equal to zero at odd multiples of π/2. Mathematically, this can be represented as: $$x=\frac{(2n+1)\pi}{2}$$ where n is an integer.

Recognize values of x where tangent function is undefined

Since division by zero is undefined, the tangent function is undefined when cos(x) = 0. Therefore, the tangent function is undefined at the values of x found in Step 2: $$x=\frac{(2n+1)\pi}{2}$$ where n is an integer.