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Chapter 4: Problem 61

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$\frac{5 \pi}{4}$$

Short Answer

Expert verified
The cosine of \(\frac{5\pi}{4}\) is \(-\frac{\sqrt{2}}{2}\), the sine of \(\frac{5\pi}{4}\) is \(-\frac{\sqrt{2}}{2}\), and the tangent of \(\frac{5\pi}{4}\) is -1.

Step by step solution

01

Understand the Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. Every angle drawn in standard position (with its vertex at the origin and its initial side on the positive x-axis) intersects the unit circle at a point. The x-coordinate of this point gives the cosine of the angle, while the y-coordinate gives the sine of the angle. Each angle on the unit circle corresponds to a real number, which is the measure of the angle in radians. One full revolution counterclockwise around the unit circle is \(2 \pi\) radians. Since we know that the angle \(\frac{5 \pi}{4}\) is slightly more than a full revolution, it should end in the third quadrant.
02

Determine the Cosine and Sine

In the third quadrant, both sine and cosine are negative. Also consider that the angle would be equivalent to \(-\frac{3 \pi}{4}\) if we subtract \(2\pi\) from it. By symmetry, these are the same as the sine and cosine of \(\frac{\pi}{4}\), but negative. Therefore, the cosine and the sine of \(\frac{5\pi}{4}\) - or equivalently \(-\frac{3\pi}{4}\) - are \(-\frac{\sqrt{2}}{2}\)
03

Compute the Tangent

By definition, the tangent of an angle is the ratio of its sine to its cosine. From step 2, we know that the sine and the cosine of our angle are both \(-\frac{\sqrt{2}}{2}\). So the tangent of \(\frac{5\pi}{4}\) is \(-\frac{\sqrt{2}/2}{\sqrt{2}/2} = -1\).

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