91Ó°ÊÓ

The secant function, denoted as \(\text{sec} \theta\), is one of the six main trigonometric functions. It is defined as the reciprocal of the cosine function. This means \(\text{sec} \theta = \frac{1}{\text{cos} \theta}\).

To better understand this, imagine a right triangle. In such a triangle, the cosine of an angle \(\theta\) is the ratio of the length of the adjacent side to the hypotenuse. Therefore:

• If \(\text{cos} \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\), then \(\text{sec} \theta = \frac{\text{hypotenuse}}{\text{adjacent}}\).

Comprehending this reciprocal relationship is crucial. By understanding \(\text{sec} \theta\), you can easily find values of \(\text{cos} \theta\) and vice versa.

The cosine function, noted as \(\text{cos} \theta\), is fundamental in trigonometry. In right-angled triangles, it gives the ratio of the length of the adjacent side to the hypotenuse.

For an acute angle \(\theta\), the cosine value will always be between 0 and 1. When we have \(\text{sec} \theta = \frac{1}{\text{cos} \theta}\), and if \(\text{sec} \theta = \frac{1}{\text{cos} \theta} \rightarrow \text{cos} \theta = \frac{1}{{\text{sec} \theta}}\), you get \(\text{cos} \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\).

In our problem, solving \(\text{sec} \theta = \frac{1}{\text{cos} \theta} = \sqrt{2} \rightarrow \text{cos} \theta = \frac{1}{\text{sec} \theta} = \frac{1}{\sqrt{2}}\), you must simplify \(\text{cos} \theta\) as: \(\text{cos} \theta = \frac{\sqrt{2}}{2}\).

Recognize this value, as it corresponds to specific angles in both degrees and radians.

To determine an acute angle \(\theta\), knowing its trigonometric function values is essential. An acute angle is between 0 and 90 degrees (<\(0 < \theta < 90^\text{\textdegree}\)).

From our problem, we found \(\text{cos} \theta = \frac{\sqrt{2}}{2}\). The next step is to determine which angle corresponds to this cosine value among acute angles.

Using trigonometric tables or unit circle knowledge, \(\text{cos} \theta = \frac{\sqrt{2}}{2}\) indicates \(\theta\) is \(45^\text{\textdegree}\) or <\(\frac{\pi}{4}\).

Thus, the acute angle \(\theta\) that satisfies our function value \(\sec \theta = \sqrt{2}\) is \(\theta = 45^\text{\textdegree}\) or <\(\frac{\pi}{4}\) radians.