Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.


<IMAGE>


sec 45°

Verified step by step guidance

1. Identify the trigonometric identity for secant: \( \sec \theta = \frac{1}{\cos \theta} \).
2. Recognize that \( \theta = 45^\circ \) is a special angle, and recall the cosine value for this angle: \( \cos 45^\circ = \frac{\sqrt{2}}{2} \).
3. Substitute the cosine value into the secant identity: \( \sec 45^\circ = \frac{1}{\frac{\sqrt{2}}{2}} \).
4. Simplify the expression by multiplying the numerator and the denominator by \( \sqrt{2} \) to rationalize the denominator: \( \frac{1}{\frac{\sqrt{2}}{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{1} \).
5. Conclude that \( \sec 45^\circ = \sqrt{2} \).

Key Concepts

Secant Function

The secant function, denoted as sec(θ), is the reciprocal of the cosine function. It is defined as sec(θ) = 1/cos(θ). Understanding this relationship is crucial for evaluating secant values, especially for common angles like 45°, where the cosine value is known.

Special Right Triangles

Special right triangles, particularly the 45°-45°-90° triangle, have specific side ratios that simplify trigonometric calculations. In a 45°-45°-90° triangle, the legs are equal, and the hypotenuse is √2 times the length of each leg. This property allows for quick evaluation of trigonometric functions at these angles.

Rationalizing the Denominator

Rationalizing the denominator is a technique used to eliminate square roots from the denominator of a fraction. This is done by multiplying the numerator and denominator by a suitable value that will result in a rational number in the denominator. This concept is important when expressing trigonometric values in a standard form.