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

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Identify the quadrant in which the angle 240° lies. Since 240° is between 180° and 270°, it lies in the third quadrant.

Find the reference angle for 240°. The reference angle is the acute angle formed with the x-axis, calculated as \(240° - 180° = 60°\).

Recall the sign of the secant function in the third quadrant. Since secant is the reciprocal of cosine, and cosine is negative in the third quadrant, secant will also be negative there.

Use the reference angle to find the exact value of \(\sec 240°\). This is \(-\sec 60°\) because of the negative sign in the third quadrant.

Recall the exact value of \(\sec 60°\). Since \(\cos 60° = \frac{1}{2}\), then \(\sec 60° = \frac{1}{\cos 60°} = 2\). Therefore, \(\sec 240° = -2\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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