Problem 103 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

When the acute angle approaches \(90^{\circ}\), the tangent of the angle gets larger and larger, approaching positive infinity.

Step 1: Understand the behaviour of the tangent function

The tangent of an angle \(\theta\) in a right triangle is defined as the ratio of the side opposite \(\theta\) to the side adjacent to \(\theta\). The tangent function is associated with the ratio of the two sides of a right triangle, and is a value that varies as the angle changes.

Step 2: Evaluate the tangent of an angle close to \(90^{\circ}\)

As the angle \(\theta\) approaches \(90^\circ\), the length of the side opposite to \(\theta\) (the hypotenuse of the right triangle) extends while the length of the side adjacent to \(\theta\) reduces, approaching zero. This implies that the ratio of these two sides, or the tangent of the angle, approaches infinity.

Step 3: Describe the behavior

As the acute angle gets close to \(90^\circ\), the value of the tangent of the angle gets larger and larger, and it actually approaches positive infinity.