Why does the graph of tangent have vertical asymptotes?

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Diagram of a right triangle labeled with sides a, b, and hypotenuse c, alongside formulas for the six trigonometric functions-sine, cosine, tangent, cosecant, secant, and cotangent-each defined by ratios of the triangle's sides. — trigonometry Based on the definitions, various simple relationships exist among the functions. For example, csc A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A, and tan A = sin A/cos A.

The graph of the tangent function has vertical asymptotes because the function is defined as the ratio of sine to cosine:

$\tan A = \frac{\sin A}{\cos A}$

In mathematics, division by zero is undefined. Therefore, the tangent function is undefined whenever the denominator, $\cos A$ , is equal to zero. These points occur at $A = \frac{π}{2} + n π$ , where $n$ is any integer. As the angle $A$ approaches these values, the value of the tangent function increases or decreases without bound, resulting in the vertical lines on the graph known as asymptotes.1