In Exercises 8–13, find the exact value of each expression. Do no...

Question

In Exercises 8–13, find the exact value of each expression. Do not use a calculator.

tan 300°

Accepted Answer

Hello everybody. I hope we're doing. All right. Today, today, I want to begin this question that states determine the exact value of the following expression without the help of the calculator where our expression is tangent of degrees. Now the answers provided are a squared of three divided by three B squared of three C negative squared of three and D negative squared of three divided by three. Now, when looking at the angle that we're working with, we can see that 240 degrees is greater than 180 degrees, but less than 270 degrees. And why do we care about this? Well, when you're looking at a coordinate system, you know that this will lie or angle will lie in quadrants three because it is greater than 180 less than 270 that area will be quadrant three. So with that, we can talk about a reference angle. Now, now in quadrants three to solve for the reference angle in or using an angle from quad and three, we can say that R angle minus 180 degrees will be equal to the reference angle. So we'll have degrees minus 180 degrees will be equal to 60 degrees. So 60 degrees will be a reference angle. And therefore, we will be working with tangents of 60 degrees. So now that we know that we have, we can look at the values that we know of the unit circle. So using the unit circle, these are all values that we should know. And we, if we look at the unit circle and we know where 60 degrees lies on, we see that this will have a point at six degrees. It will be a point of one half comma squared of three divided by two. And we should also know that with the unit circle, the X component is equal to cosine of theta and the Y component is equal to sign of theta. Therefore, if we have tangents of 60 degrees, this will be equal to sine of 60 degrees divided by the cosine of six degrees because tangent theta is equal to sine of theta divided by cosine theta. And as we mentioned, the sign of theta is equal to the Y component of that data. And if we look at that Y component of that point, we have squared of three divided by two and that'll be divided by the close of theta. And so we just look at the X component of that point which is one half. So we have squared of three divided by two divided by one half. So we'll have that tangent of 240 degrees is equal to the tangent of 60 degrees. And this will be equal to, if we have squared of three divided by two, divided by one half, the twos will cancel out. And we're left with squared of three divided by one which is square roots of three. And that will be our final answer which matches with Andrew choice B. So as you can see, knowing the different values of the unit circle is very important when it comes to trigonometric functions as well as their relationships. And knowing things like that allow us to solve questions like this one. So I hope that was helpful. And until next time.

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In Exercises 8–13, find the exact value of each expression. Do not use a calculator. tan 300°