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Chapter 10: Problem 62

Describe what happens to the tangent of an angle as the measure of the angle gets close to \(90^{\circ}\).

Short Answer

Expert verified
As the measure of an angle approaches \(90^{\circ}\), the tangent of the angle becomes infinitely large.

Step by step solution

01

Understand the Tangent Function

The tangent of an angle \( \theta \) in a right triangle is defined as the ratio of the length of the side opposite to \( \theta \) over the length of the side adjacent to \( \theta \) or \( \tan ( \theta ) = \frac{\text{opposite}}{\text{adjacent}} \).
02

Analyze the behavior of tangent as the angle increases

As the angle \( \theta \) approaches \(90^{\circ}\) (or \( \frac{\pi}{2} \) in radians), the length of the side opposite to \( \theta \) gets longer while the length of the side adjacent to \( \theta \) gets shorter. This makes the ratio \( \frac{\text{opposite}}{\text{adjacent}} \) increase without bound.
03

Draw Conclusion

Therefore, as the angle \( \theta \) gets closer to \(90^{\circ}\), the tangent of the angle, \( \tan ( \theta ) \), gets infinitely large, that is, \( \tan ( \theta ) \rightarrow \infty \).

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