Give the exact value of each expression. See Example 5. 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 exact value of \(\cos 45^\circ\). From the unit circle or special triangles, \(\cos 45^\circ = \frac{\sqrt{2}}{2}\).
4. Substitute the value of \(\cos 45^\circ\) into the secant formula: \(\sec 45^\circ = \frac{1}{\frac{\sqrt{2}}{2}}\).
5. Simplify the expression by multiplying numerator and denominator appropriately to find the exact value of \(\sec 45^\circ\).

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 is important to understand this relationship to find the exact value of secant for a given angle.

Exact Values of Trigonometric Functions for Special Angles

Certain angles like 30°, 45°, and 60° have well-known exact trigonometric values. For 45°, cos(45°) = √2/2, which helps in calculating sec(45°) precisely without a calculator.

Simplifying Radical Expressions

After finding the reciprocal of cosine, simplifying the resulting expression, often involving square roots, is necessary to express the answer in its simplest exact form, such as rationalizing denominators.