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Chapter 4: Problem 16

evaluate (if possible) the sine, cosine, and tangent at the real number. $$t=-\frac{\pi}{4}$$

Short Answer

Expert verified
\(\sin(-\frac{\pi}{4}) = -\sqrt{2}/2, \cos(-\frac{\pi}{4}) = \sqrt{2}/2, \tan(-\frac{\pi}{4}) = -1\)

Step by step solution

01

Evaluate Sine Function

To find the sine, use the unit circle definition of sine, which is the y-coordinate of the point where the line through the origin forms an angle of \(t\) with the positive x-axis. For \(t=-\frac{\pi}{4}\), we are in the fourth quadrant where sine is negative. The angle \(\frac{\pi}{4}\) has a known sine value of \(1/\sqrt{2}\), hence the sine of \(-\frac{\pi}{4}\) is -\(1/\sqrt{2}\) or -\(\sqrt{2}/2\).
02

Evaluate Cosine Function

The cosine function is the x-coordinate of the point on the unit circle. For \(t=-\frac{\pi}{4}\), we are still in the fourth quadrant where cosine is positive. The cosine of \(\frac{\pi}{4}\) is \(1/\sqrt{2}\), hence the cosine of \(-\frac{\pi}{4}\) is also \(1/\sqrt{2}\) or \(\sqrt{2}/2\).
03

Evaluate Tangent Function

The tangent of an angle in the unit circle is the y-coordinate divided by the x-coordinate. So, \(\tan(-\frac{\pi}{4}) = \frac{\sin(-\frac{\pi}{4})}{\cos(-\frac{\pi}{4})} = \frac{-\sqrt{2}/2}{\sqrt{2}/2} = -1\).

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