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


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sec 45°

Verified step by step guidance

1. Recall the definition of secant in terms of cosine: \(\sec \theta = \frac{1}{\cos \theta}\).
2. Identify the angle given: \(45^\circ\).
3. Find the value of \(\cos 45^\circ\). From the special right triangle (45°-45°-90°), \(\cos 45^\circ = \frac{\sqrt{2}}{2}\).
4. Substitute this value into the secant formula: \(\sec 45^\circ = \frac{1}{\frac{\sqrt{2}}{2}}\).
5. Rationalize the denominator by multiplying numerator and denominator by \(\sqrt{2}\) to eliminate the square root in the denominator.

Key Concepts

Definition of Secant Function

The secant function, sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). It relates the hypotenuse to the adjacent side in a right triangle and is used to find ratios involving angles.

Exact Values of Trigonometric Functions for Special Angles

Certain angles like 45° have well-known exact trigonometric values. For 45°, cos(45°) = √2/2, so sec(45°) = 1/(√2/2) = √2. Knowing these values helps evaluate expressions without a calculator.

Rationalizing the Denominator

Rationalizing the denominator involves eliminating square roots from the denominator of a fraction by multiplying numerator and denominator by a suitable radical. This simplifies expressions and is often required for final answers.