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 by the terminal side of an angle in standard position and the x-axis. It is always measured as a positive angle and is used to simplify the calculation of trigonometric functions. For angles greater than 180°, the reference angle can be found by subtracting 180° from the angle, which helps in determining the values of trigonometric functions in different quadrants.

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 the secant function is crucial for evaluating expressions involving angles, especially when determining values in different quadrants, as the sign of the cosine function will affect the secant's value.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a fundamental tool in trigonometry, as it provides a geometric interpretation of the sine, cosine, and tangent functions. By using the unit circle, one can easily find the coordinates of points corresponding to specific angles, which are essential for calculating trigonometric values like sec(240°).