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 80° - tan(-55°)]/[ 1 + tan 80° tan(-55°)]

Accepted Answer

Hey, everyone in this problem, we're asked to determine the exact value of the following trigonometric expression. And the expression we have is tangent of 100 degrees minus tangent of negative 50 degrees divided by one plus tangent of 100 degrees multiplied by tangent of negative 50 degrees. We have four answer choices. Option A negative square root of three. Option B the square root of three, option C negative square root of three divided by three and option D the square root of three divided by three. Now, if we take a look at this expression, OK, tangent of 100 degrees tangent of negative 50 degrees, these are not nice angles that we can evaluate tangent at. OK. That we can get an exact value for. So we can't just go through and evaluate this piece by piece as it is. And we need to use some trig identities to transform this expression that we have into something that's gonna give us some nicer exact values. So we're call that if we have tangent of A minus B, we can write this as tangent of A minus tangent of V divided by one plus tangent of a tangent of V hm. Now, the expression on the right hand side matches exactly with the expression we were given where A is going to be 100. OK. And B is going to be negative 50. So we can write this big expression, we were given as tangent of A minus B where A is equal to 100 degrees and B is equal to negative 50 degrees. All right. So let's go ahead and do that. We have tangent of bay minus tangent of B oops, sorry. And let's fill in our values here, tangent of 100 degrees minus tangent of negative 50 degrees. All divided by one plus tangent of 100 degrees multiplied by tangent of negative 50 degrees. This is going to be equal to tangent of 100 degrees minus negative 50 degrees. OK? 100 degrees minus negative 50 degrees minus A negative is gonna give us plus. So we get tangent of 100 and 50 degrees. Now 150 degrees we can think about our unit circle and determine the value of tangent at 150 degrees. OK. Another thing we can do is draw this out and think about the reference angle. OK. So if we draw this out 150 degrees is gonna be in quadrant two. OK. So from the positive X axis counterclockwise, we measure 150 degrees. That's the angle we're interested in, but we can look at the reference angle which is gonna be the angle to the negative X axis. We call this the prime. Now theta prime is going to be what? Well, it's gonna be 180 degrees minus 150 degrees. And we know from the positive X axis all the way to the negative X axis is 180 degrees and it makes a straight line and then we're subtracting that 150 degree angle that we already have a, the two of these together make that 180 degrees. So when we do that, we get a reference angle of 30 degrees. So instead of evaluating tangent of 100 and 50 degrees, we can evaluate tangent of 30 degrees. We have to be a little bit careful though we have to consider the sign of tangent. Now we're in quadrant two, which means a tangent will be negative. OK. So tangent is negative in quadrant two. OK. It's really important to remember which quadrants you have positive and negative values for sine cosine tangent. OK. So what that means is tangent of 1 5100 and 50 degrees is not going to be the same as tangent of 30 degrees. It's going to be negative tangent of 30 degrees. OK. So we can evaluate it at that reference angle, but we need that negative in there. Now, tangent of 30 degrees, this is hopefully a value that you've remembered, OK, you can always do tangent as sine divided by cosine as well. So if you know sine of 30 degrees and cosine of 30 degrees, you can do it that way as well. And what we're gonna have is negative one divided by the square root of three. We can rationalize the denominator. So we're gonna multiply numerator and denominator by the square root of three to get rid of that square root and the denominator. And so we end up with negative square root of three divided by three. So we started with this big messy expression with multiple tangents. And what we found is that the exact value is just negative the square root of three divided by three. If we compare this to the answer traces, we found we can see that it corresponds with option C. Thanks everyone for watching. I hope this video helped you in the next one.

Find the exact value of each expression. See Example 1.
[tan 80° - tan(-55°)]/[ 1 + tan 80° tan(-55°)]

Key Concepts

Tangent Function and Its Properties

Tangent Addition Formula

Recognizing the Expression as a Tangent Difference

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