In Exercises 29–51, find the exact value of each expression. Do n...

Question

In Exercises 29–51, find the exact value of each expression. Do not use a calculator.
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csc(tan⁻¹ √3/3)

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the expression. The C can of the arc tangent of radical three without using a calculator, we have four answer choices. First, I am going to rewrite the arc tangent of radical three and set it equal to alpha which represents any angle. We don't know which one yet from there. I know that I can take the tangent of both sides. And so I'll have the tangent of alpha equals radical three. Since we're not using a calculator and we do need to use a unit circle. I'm gonna break tangent into the sign of alpha divided by the cosine of alpha, which means this would be radical three divided by one. So the sign would be radical three, the cosine would be one. However, if you notice in our unit circle, all of our values are divided by two. So there's a good possibility that this is radical three divided by two, divided by one, divided by two because if this had been the case, we would have canceled out the twos and the denominators. So this seems to make sense. So we have that the sign of alpha is radical three divided by two. The cosine of alpha is one half from here. When I look at my unit circle, the place where cosine is one half and sine is radical three divided by two is at the angle pi divided by three. This also could happen at the angle four pi divided by three. However, tangent or sorry, the arc tangent has to be within the range of negative pi divided by two to pi divided by two. So this cannot equal four pi divided by three, our alpha must be pi divided by three. So now that we know that alpha equals pi divided by three, this means that the arc tangent of radical three equals pi divided by three. So we're gonna substitute that in. So we're gonna do the C can of pi divided by three. Recall that C can is the same as one divided by the cosine. So this will be one divided by the cosine of pi divided by three, one divided by and then the cosine of pi divided by three is one half. So when I clean up my um complex fraction, I have one divided by one half, which is the same as saying one multiplied by the reciprocal of two, one multiplied by two is two. So this means that the sea can't of the arc tangent of radical three equals two. And that's answer choice. D have a nice day.

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In Exercises 29–51, find the exact value of each expression. Do not use a calculator. _ csc(tan⁻¹ √3/3)