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Chapter 4: Problem 54

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$300^{\circ}$$

Short Answer

Expert verified
\(\sin 300掳 =- \frac{\sqrt{3}}{2}\), \(\cos 300掳 = \frac{1}{2}\), and \(\tan 300掳 = -\sqrt{3}\)

Step by step solution

01

Convert degrees to radians

As a first step, it's easier to work with radians than with degrees in trigonometry, so we should convert 300 degrees into radians. The formula to do this is to multiply the degrees by \( \frac{\pi}{180} \). So, \(300 \times \frac{\pi}{180} = \frac{5\pi}{3}\) radians.
02

Find the reference angle

A reference angle is the acute version of the angle we鈥檙e dealing with, formed by the terminal side of the angle and the nearest axis. For an angle in standard position, its terminal side lies along 300掳 in the counterclockwise direction. As a result, it's in the fourth quadrant, and we can find the reference angle by subtracting 300掳 from 360掳. The reference angle then is 60掳 or \( \frac{\pi}{3} \) radians.
03

Find the sine, cosine, and tangent of the reference angle

We reduced our problem to finding sine, cosine, and tangent for 60掳 or \( \frac{\pi}{3} \) radians. As \( \sin 60掳 = \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \), \( \cos 60掳 = \cos \frac{\pi}{3} = \frac{1}{2} \), and \( \tan 60掳 = \tan \frac{\pi}{3} = \sqrt{3} \).
04

Determine the signs

In the fourth quadrant, where our original angle is, only cosine is positive and both sine and tangent are negative. So, we should multiply each of our obtained values by -1, except for cosine.
05

Present the final values

We now have all the values we need, just replace the values of sine, cosine, and tangent from step 4, taking into account the signs.

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