Find exact values of the six trigonometric functions for each ang...

Question

Find exact values of the six trigonometric functions for each angle. Do not use a calculator. Rationalize denominators when applicable.

120°

Accepted Answer

Welcome back. I am so glad you're here. We're asked to list the six trigonometric function values of the following angle provide exact values when necessary, rationalize the denominator. Our angle is 150 degrees. So first let's figure out where 150 degrees is, we can draw a rough sketch of a coordinate plane. We have a vertical Y axis and then it's supposed to be a horizontal X axis though minus a little curvy for 150 degrees that has its vertex at the origin, its initial side along the positive part of the X axis and its terminal side goes all the way into the second quadrant as it goes counterclockwise, it's heading toward negative infinity for the values and positive infinity for the Y values a little bit closer to the negative part of the X axis than the positive part of the Y axis. Now to figure out the reference angle for our quadrant two angle, we're going to take 180 degrees minus our angle which is 150 degrees and that gives us 30 degrees. So our reference angle is 30 degrees and we recall from previous lessons that we have exact values for the six trigonometric functions of 30 degrees right in a chart in the textbook. So I'm going to draw part of that chart. So we'll have a column for the sign of the, we'll have a column for the cosine of theta, a column for the tangent of theta, a column for the cotangent of theta, a column for the, of the, and a column for the Koi can of theta. And there's our six trigonometric functions. That's the first row. And then we'll have two rows beneath the first row will be right from the textbook. And from previous lessons, that's going to be our 30 degrees. But 30 degrees is over in quadrant one where all of our trigonometric functions are positive degrees. The one we're finding is in quadrant two and not everything is positive there. Although the exact values will be the same for their absolute value. All right. So let's first fill in for 30 degrees. So the sign of 30 degrees is one half, the cosine of 30 degrees is the square of three divided by two. The tangent of 30 degrees is the square of three divided by three. The cotangent of 30 degrees is the square of three. The sea can of 30 degrees is two square at three divided by three and the can of 30 degrees is two. It's lovely that they have already rationalized these denominators for us now for 150 degrees, it's going to be the same values but some will be positive and some will be negative. So what's happening in the second quadrant, what's positive and what's negative there? Well, we recall from the pneumonic device, all students take calculus and that in quadrant two, the S stands for sine. So sign is positive there as well as its reciprocal function coy can. So those two are going to be the positive ones. All the rest are going to be negative cosine is going to be negative tangent will be negative cotangent will be negative and C can will be negative in quadrant two. So we can just bring down all of these other things. So for 150 degrees, the sign is a positive one half for 150 degrees. The cosine is a negative square at three divided by two. The tangent of 150 degrees is a negative square at three divided by three. The cotangent of 150 degrees is a negative square at three. The C can of 150 degrees is a negative two square at three divided by three. And the CO C can of 150 degrees is still a positive two. And there is our solution. Well done. We'll catch you on the next one.

Find exact values of the six trigonometric functions for each angle. Do not use a calculator. Rationalize denominators when applicable. 120°

Key Concepts

Unit Circle

Reference Angles

Trigonometric Function Values

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