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Chapter 4: Problem 60

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$\frac{3 \pi}{4}$$

Short Answer

Expert verified
The sine of \(\frac{3 \pi}{4}\) is \(\frac{\sqrt{2}}{2}\), cosine is \(-\frac{\sqrt{2}}{2}\), and tangent is -1.

Step by step solution

01

Sine

Starting with sine, recall that sine of an angle in the second quadrant is positive. The reference angle for \(\frac{3 \pi}{4}\) is \(\frac{\pi}{4}\), and \(\sin(\frac{\pi}{4})\) equals \(\frac{\sqrt{2}}{2}\). Therefore, \(\sin(\frac{3 \pi}{4})\) is also \(\frac{\sqrt{2}}{2}\).
02

Cosine

Next, for cosine, remember that cosine of an angle in the second quadrant is negative. Once again, the reference angle is \(\frac{\pi}{4}\), so \(\cos(\frac{\pi}{4})\) equals \(\frac{\sqrt{2}}{2}\). As such, \(\cos(\frac{3 \pi}{4})\) will be \(-\frac{\sqrt{2}}{2}\).
03

Tangent

Finally, to find the tangent, we take the sine of the angle and divide it by the cosine of the angle. So, \(\tan(\frac{3 \pi}{4})\) equals \(\frac{\sin(\frac{3 \pi}{4})}{\cos(\frac{3 \pi}{4})}\). Therefore, \(\tan(\frac{3 \pi}{4}) = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = -1\).

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