In Exercises 49–59, find the exact value of each expression. Do n...

Question

In Exercises 49–59, find the exact value of each expression. Do not use a calculator.

tan 120°

Accepted Answer

Hello, today we're gonna be finding the value of the given expression. So what we are given is tangent of 150 degrees. So one way to approach this problem is to use reference angles. But in order to get a reference angle, we need to find the location of 150 degrees on the unit circle. So where is 150 degrees located on the unit circle? While 150 degrees is going to be located in the second quadrant of the unit circle. And now that we know its location, we can go ahead and calculate the following reference angle. Now, the reference angle is located between our given angle and the x axis. And in order to get a reference angle in quadrant two, we're going to take the measure of 100 and 80 degrees and subtract our given angle which is 150 degrees, 180 minus 150 gives us degrees. So what this means is that the reference angle has a measure of degrees. Now that we have the reference angle, we can use this angle to create a right, a, a right triangle in the second quadrant. And this allows us to state that tangent of 150 degrees is going to be equivalent to tangent of 30 degrees of the following right triangle. So let's go ahead and draw a bigger version of this right triangle that is located in the second quadrant. So again, this is a right triangle. So we have a right angle located here, we know that the reference angle has a measure of 30 degrees. And so now what we wanna do is we want to solve for tangent of 30 for the following right triangle. Now this is possible by using their special right triangles. And in this case here, we're going to be using a 30 60 90 degree triangle across from 30 degrees is going to be a length of one across from 60 degrees is going to be a length of the square root of three and across from 90 degrees is going to be a length of positive two. Now, one thing to note is that this right triangle is in the second quadrant, any X value in the second quadrant or in this case, the length of the triangle is going to be negative and any Y value or the height of the triangle is going to be positive in this quadrant. So what this means is that the length of the right triangle is going to be negative. So we have negative square root of three and the height is positive. So it stays as positive one. So how do we get tangent of 30 degrees for the following right triangle? Well, recall the tangent for of any right triangle is defined as our Y value the height of the triangle over the X value which is the length of the triangle. So our Y value in this case is going to be positive one and our X value is going to be negative square root of three. What this means is that tangent of 30 degrees is going to be one over negative square root of three. Now, we cannot leave a square root in our solution. In the denominator, we're going to have to rationalize this quantity in order to do that, we're going to multiply the numerator and denominator by the square root of three. And this is going to simplify the quantity to give us negative square root of 3/3. So what this means is that tangent of 30 degrees of the following right triangle is equal to negative square root 3/3. And recall that earlier, we stated that tangent of 150 degrees is equivalent to the measure of tangent of 30 degrees of the following right triangle. What this means is that tangent of 150 degrees is going to have the value of negative square root 3/3. And this is going to be the value for the given expression. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. tan 120°

Key Concepts

Unit Circle

Tangent Function

Reference Angles

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