Problem 106 - Algebra and Trigonometry

Question

Explain why tan \(90^{\circ}\) is undefined.

Step-by-Step Solution

Answer

The tangent of \(90^{\circ}\) is undefined because it results in a division by zero scenario, which is undefined in mathematics.

Step 1: Understanding the Tangent Function

The tangent function, abbreviated as tan, in trigonometry is the ratio of the sine to the cosine of an angle. That is, \(\text{tan} (\theta) = \frac{\text{sin} (\theta)}{\text{cos} (\theta)}\), where \(\theta\) represents an angle.

Step 2: Evaluation of Sine and Cosine

Firstly, evaluate the sine and cosine for \(90^{\circ}\). From the unit circle, we find that \(\text{sin} (90^{\circ}) = 1\) and \(\text{cos} (90^{\circ}) = 0\).

Step 3: Substitution into the Tangent Function

Substitute these values into the formula for the tangent, \(\text{tan} (\theta) = \frac{\text{sin} (\theta)}{\text{cos} (\theta)}\). Resulting in: \(\text{tan} (90^{\circ}) = \frac{1}{0}\).

Step 4: Explanation of the Undefined Result

Division by zero is undefined in mathematics because there is no number that can multiply by zero to give a non-zero number. Hence the tangent of \(90^{\circ}\) is undefined.