Find the exact value of each expression. See Example 1.
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Question

Find the exact value of each expression. See Example 1.

[tan 5π/12 + tan π/4]/[1 - tan 5π/12 tan π/4]

Accepted Answer

Hey, everyone. Welcome back in this problem. We're asked to determine the exact value of the following trigonometric expression. We're giving tangent of 11 pi divided by 12 plus tangent of pi divided by four, all divided by one minus tangent of 11 pi divided by 12, multiplied by tangent of pi divided by four. We have four answer choices. Option A negative square root of three divided by three. Option B square root of three. Option C, the square root of three divided by three and option D negative the square root of three. So we wanna look at this expression we were given and we wanna think about our trigonometric identities. Are there any identities we know that give us a result similar to this? OK. That we could kind of manipulate to work for us. And there is OK. Recall that if we have tangent of A plus B, we can write this as tangent of A plus tangent of B divided by one minus tangent of a tangent of the bee. Now this expression we've written on the right hand side is exactly the same as the expression we were given in the question. OK. Where A value is going to be 11 pi divided by 12 and our B value is going to be pi divided by four. OK. All right. So we can write our original expression tangent of 11 pi divided by 12 mm plus tangent of pi divided by four, divided by one minus tangent. 11 pi divided by 12, multiplied by tangent of pi divided by four as tangent of 11 pi divided by 12 plus pi divided by four, right. So we're working backwards. We started with the expression on the right hand side, we've written it as the simplified expression on the left hand side, tangent of A plus B. OK. They are a value an RV. Now we've simplified it down to a single trigonometric function, a single tangent function and we just need to simplify within the brackets. So let's do that. We're gonna start by finding a common denominator. So we can write this as tangent of 11 pi divided by 12 plus multiplying by three in the numerator and denominator three pi divided by 12. Let's keep going. This gives us tangent 14 pie divided by 12. And we can simplify dividing numerator and denominator by 22 tangent of seven pi divided by six. Now let's recall that if we have tangent of some angle theta plus pi, this is equivalent to just tangent of theta. OK? Because the period of tangent is pi. So if we have tangent of seven pi divided by six, we can say that this is some angle theta plus pi. OK. That's gonna be our seven pi divided by six. And so our theta value is gonna be seven pi divided by six minus pi which is just equivalent to pi divided by six. OK. So tangent of seven pi divided by six is going to be equal to tangent of pi divided by six. We just subtracted pi from that original angle to get us down to an angle where we know the value. OK. So we're finding an equivalent angle. OK. Now that we have tangent of pi divided by six, we can do this in one of two ways. If you remember the value of tangent at pi divided by six, you can go ahead and use that right away. If not, you can break tangent up into sine and cosine. And we're gonna do that to show you how to do this just in case you've forgotten. OK. So we can break tangent up into sine of pi divided by six, divided by cosine of pi divided by six. And these values of sine and cosine are values that you should know or values that you can use your um triangle identities to find. OK. So sine of pi divided by six. Well, this is equal to one half cosine of pi divided by six is equal to the square root of three, divided by two. Now our factors of two are gonna divide out, we're gonna be left with one divided by the square root of three. And then we just want to rationalize the denominator. We don't want these square roots in the denominator. So we're gonna multiply the numerator and denominator by the square root of three. And we get that our function or our expression is going to be equal to the square root of three divided by three. All right. So that looked messy. We started with this really messy expression with a bunch of tangents. We were able to use our addition identity to simplify it into a single tangent function. Once we did that, it was just a matter of simplifying within the brackets and then determining the value associated with tangent of pi divided by six. If we go to our answer choices, we can see that the answer we found corresponds with option C A, the square root of three divided by three. Thanks everyone for watching. I hope this video helped see you in the next one.

Find the exact value of each expression. See Example 1.
[tan 5π/12 + tan π/4]/[1 - tan 5π/12 tan π/4]

Key Concepts

Tangent Addition Formula

Exact Values of Special Angles

Angle Conversion and Decomposition

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