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Chapter 27: Problem 849

Find the values of the trigonometric functions of an angle of \(-510^{\circ}\).

Short Answer

Expert verified
For the angle -510掳 or 210掳, the values of the trigonometric functions are: \(\sin(-510^\circ) = -\frac{1}{2}\) \(\cos(-510^\circ) = -\frac{\sqrt{3}}{2}\) \(\tan(-510^\circ) = \frac{\sqrt{3}}{3}\)

Step by step solution

01

Convert the angle to the range of 0掳 to 360掳

To find the corresponding positive angle in the range of 0掳 to 360掳, we will add 360掳 to the given angle (-510掳) until we get an angle in the required range. \(- 510^\circ + 360^\circ = -150^\circ\) \(-150^\circ + 360^\circ = 210^\circ\) So, the angle of 210掳 corresponds to the angle of -510掳 in the range of 0掳 to 360掳.
02

Determine the quadrant of the angle

Now we need to determine the quadrant of the 210掳 angle. Since the angle is more than 180掳 and less than 270掳, we can say that the angle lies in the third quadrant.
03

Find the reference angle

To find the reference angle, we will subtract 180掳 from the angle. \(210^\circ - 180^\circ = 30^\circ\) The reference angle of the 210掳 angle is 30掳.
04

Find the values of the trigonometric functions of the reference angle

Now, we will find the values of sine, cosine, and tangent for the reference angle of 30掳. For 30掳, the values of the trigonometric functions are as follows: \(\sin(30^\circ) = \frac{1}{2}\) \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\) \(\tan(30^\circ) = \frac{\sqrt{3}}{3}\)
05

Determine the trigonometric function values for the original angle using the reference angle values

As the given angle lies in the third quadrant, both sine and cosine are negative, and tangent is positive. Therefore, we will use the values of sine, cosine, and tangent for the reference angle, but with appropriate signs. For the angle -510掳 or 210掳, the values of the trigonometric functions are: \(\sin(-510^\circ) = \sin(210^\circ) = -\frac{1}{2}\) \(\cos(-510^\circ) = \cos(210^\circ) = -\frac{\sqrt{3}}{2}\) \(\tan(-510^\circ) = \tan(210^\circ) = \frac{\sqrt{3}}{3}\) So, the values of the trigonometric functions of the angle -510掳 are: sin(-510掳) = -1/2, cos(-510掳) = -鈭�3/2, tan(-510掳) = 鈭�3/3.

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