Problem 64 - Precalculus Essentials

Question

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\sec 240^{\circ}$$

Step-by-Step Solution

Answer

The exact value of \( \sec 240^{\circ} \) is \(-2\).

Step 1: Finding the Reference Angle

A reference angle is the acute angle made by the terminal side of any angle in standard position with the x-axis. To find the reference angle for \(240^{\circ}\), we subtract it from \(360^{\circ}\) if it allows us to land in the first quadrant. However, since \(240^{\circ}\) is smaller than \(360^{\circ}\), we subtract \(180^{\circ}\) from it instead. So, the reference angle is \(240^{\circ} - 180^{\circ} = 60^{\circ}\).

Step 2: Identifying the Quadrant

Any angle greater than \(180^{\circ}\) but less than \(360^{\circ}\) lies in the third quadrant, therefore, \(240^{\circ}\) is in the third quadrant. In this quadrant, the secant function is negative. This is important because the sign of the trigonometric function depends on the quadrant in which the original angle lies.

Step 3: Using the Trigonometric Identity

The secant function is the reciprocal of the cosine function. This means we can change our initial expression \(\sec 60^{\circ}\) to \(1/\cos 60^{\circ}\). As \(\cos 60^{\circ}\) equals \(1/2\), then the secant equals \(1/(1/2) = 2\). Thus, \(\sec 240^{\circ} \) equals \(-2\) because the secant function is negative in the third quadrant.