In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. sec 240°

Verified step by step guidance

1. Identify the reference angle for 240°. Since 240° is in the third quadrant, subtract 180° from 240° to find the reference angle: 240° - 180° = 60°.
2. Recall that the secant function, \( \sec \theta \), is the reciprocal of the cosine function, \( \cos \theta \). Therefore, \( \sec \theta = \frac{1}{\cos \theta} \).
3. Determine the sign of the secant function in the third quadrant. In the third quadrant, cosine is negative, so secant will also be negative.
4. Use the reference angle to find \( \cos 60° \). From trigonometric tables or knowledge, \( \cos 60° = \frac{1}{2} \).
5. Calculate \( \sec 240° \) using the reference angle: \( \sec 240° = \frac{1}{\cos 240°} = \frac{1}{-\frac{1}{2}} = -2 \).

Key Concepts

Reference Angles

A reference angle is the acute angle formed between the terminal side of a given angle and the x-axis. It helps simplify trigonometric calculations by relating angles in different quadrants to their acute counterparts, allowing the use of known values for sine, cosine, and other functions.

Secant Function and Its Relationship to Cosine

The secant function, sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). Understanding this relationship allows you to find secant values by first determining the cosine of the angle, especially useful when working with exact values.

Trigonometric Values in Different Quadrants

The sign of trigonometric functions depends on the quadrant in which the angle lies. For 240°, located in the third quadrant, cosine (and thus secant) is negative. Recognizing quadrant signs is essential to assign the correct sign to the exact trigonometric value.