Find the exact values of (a) sin s, (b) cos s, and (c) tan s f...

Question

Find the exact values of (a) sin s, (b) cos s, and (c) tan s for each real number s. See Example 1.

s = 2π

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine the exact values of the sine of P, the cosine of P in the tangent of P where P is equal to six pi. Our answer choices are answer choice. A, the sine of six pi equals one, the cosine of six pi equals zero. And the tangent of six pi is undefined answer choice B the sine of six pi equals negative one, the cosine of six pi equals zero. And the tangent of six pi equals undefined answer choice C the sine of six pi equals zero. The cosine of six pi equals negative one. And the tangent of six pi equals zero. And answer choice D the sine of six pi equals zero. The cosine of six pi equals one and the tangent of six pi equals zero. All right. So we've got this angle of six pi and it would be a little bit easier for us if our angle P was in between zero and two pie, then we can find a co terminal angle for six pi that's between zero and two pi, we'll just subtract one full rotation until we get to that range. So from six pi, we're going to subtract two pi that equals four pi not there yet. We'll subtract another two pie and four pi minus two pi equals two pi. So we are right at two pi. Well, we recall from previous lessons and from the figure of the unit circle in the textbook that two pi is a quadrant angle. I'll draw a really rough sketch of the unit circle. Definitely check out that figure in the textbook. For a more detailed one, we have a vertical y axis and a horizontal x axis. They come together the origin in the middle, the unit circle lies over that every point on the unit circle is exactly one unit away from the origin. That's not true of what I drew. But you can use your imagination. Now for two pi that one is right on the positive part of the X axis, that's the same as this initial side. We are right where the unit circle intercepts the positive part of the X axis. And because this is a unit circle, that's exactly one unit away. Our X value is one. And we can see that our Y value is zero. Why do we need to know the X and Y values? Because we recall from previous lessons that when we're talking about angles on the unit circle, the sign of an angle on the unit circle is going to be equal to its Y value the cosine is going to be equal to the X value and the tangent is going to be equal to Y divided by X. So for the sine of P, that's equal to Y, the Y value here is zero. So our sine of P equals zero for the cosine of P that equals X and X here is one, the cosine of P equals one and the tangent will take a second longer. We have Y divided by XY is zero, divided by X which is 10, divided by one is zero. So the tangent of P is zero. So when P equals six pi our sign is zero, our cosine is one and our tangent is zero. Looking at our answer choices that matches with answer choice. D well done. We'll catch you on the next one.

Find the exact values of (a) sin s, (b) cos s, and (c) tan s for each real number s. See Example 1.
s = 2π

Key Concepts

Unit Circle

Trigonometric Functions

Periodic Nature of Trigonometric Functions

Watch next