In Exercises 63–68, find the exact value of each expression. Do n...

Question

In Exercises 63–68, find the exact value of each expression. Do not use a calculator.

tan 𝜋/3 - ___1___ 
2 sec 𝜋/6

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the following expression without the help of a calculator. That expression is the tangent of pi divided by six, all of which is divided by three minus one divided by the C can of pi divided by three. Our answer choices are a radical three divided by three minus one half. B radical three divided by three plus one half, C radical three divided by nine minus one half D radical three divided by nine plus one half. So first, I'm gonna rewrite this and I have again, the tangent of pi divided by six is my numerator. So then it's divided by three minus one divided by the C can of pi divided by three. So I'm gonna start with, what do I know how can I use a unit circle to help me? So the first thing I know is that the tangent of pi divided by six is the same thing as the sign of pi divided by six divided by the cosine of pi divided by six. Looking at a unit circle, I know that the sign of pi divided by six is one half and the cosine of pi divided by six is radical three divided by two. So I can make this a complex fraction. And I'd have one half divided by radical three divided by two. If I want to rewrite this, I'd have one half divided by radical three divided by two using my fraction rules. I know that's the same as one half multiplied by two divided by radical three cross reducing. I get one divided by radical three. So that is what the tangent of pi divided by six will be one divided by radical three. I want to see if I can break down the sea can also before I plug this back in. So I recall that sea can of pi divided by three will be the same as one divided by the cosine of pi divided by three. And that the cosine of pi divided by three when I reference a unit circle equals one half. So this means the CK is one divided by one half to the same as one multiplied by its reciprocal. So that's 2/1, which is two. So the Sea Can of Pi divided by three is two. I'm going to plug those values. I just found to my original expression. So we just found that the tangent of pi divided by six is one divided by radical three. So we have one divided by radical three, divided by three minus one, divided by the sea can of pi divided by three, which we just said is two. So we have one divided by radical three, then divided by three minus one half. So I need to make that first fraction a little less messy. And then we should have our answer. So one divided by radical three, divided by three is the same as saying one divided by radical three, multiplied by the reciprocal which is one third. So we'd have one divided by three radical three. I do want to rationalize that denominator. So I'm gonna multiply numerator and denominator by radical three. And when I do so I get radical three divided by well radical three multiplied by radical three is three, three multiplied by three is nine. So my first fraction works out to radical three divided by nine minus the second fraction which is one half which can stay one half. It looks like they don't want us to combine it into one complicated looking fraction. So we're gonna leave it as is so radical three divided by nine minus one half is our answer. And that matches answer. Choice C have a nice day.

In Exercises 63–68, find the exact value of each expression. Do not use a calculator. tan 𝜋/3 - _1_ 2 sec 𝜋/6

Key Concepts

Tangent Function

Secant Function

Exact Values of Trigonometric Functions

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