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Chapter 4: Problem 68

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$-\frac{23 \pi}{4}$$

Short Answer

Expert verified
The sine, cosine, and tangent of the angle \(-\frac{23 \pi}{4}\) are \(\frac{\sqrt{2}}{2}\), \(\frac{\sqrt{2}}{2}\), and 1, respectively.

Step by step solution

01

Normalize the angle

Normalize the angle \(-\frac{23 \pi}{4}\) by adding multiples of \(2\pi\) until we get an equivalent angle within the range \([0, 2\pi]\). We calculate this as \(-\frac{23 \pi}{4} + 6 * 2\pi = \frac{\pi}{4}\), where 6*2\pi is added because \(2\pi\) is the period of sine and cosine, so adding any multiple of \(2\pi\) results in the same values of these functions.
02

Compute the sine, cosine and tangent

Now, compute the sine, cosine and tangent of \(\frac{\pi}{4}\). From the standard unit circle in trigonometry, we know that \(\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\), \(\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\), and \(\tan(\frac{\pi}{4}) = 1\) since \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).

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