Find the exact value of each expression. See Example 3.tan(-1020°...

Question

Find the exact value of each expression. See Example 3.tan(-1020°)

Accepted Answer

Welcome back. Everyone. In this problem, we want to determine the exact value of the trigonometric expression tangent of negative 1320 degrees. For our answer choices A is the negative skirt of three divided by three B is the skirt of three divided by three C is the square root of three and D is the negative square root of three. Now, we want to find the exact value of our trigonometric expression. And that means that we'll need to find a core terminal angle for the tangent of negative 1320 because that angle is not on our typical unit circle, right. It's our unit circle ranges between 0 to 360 degrees. So if I were to draw it, what we're really looking for is that if we had that anger negative 1320 degrees, we're looking for a core terminal angle for that the least positive core terminal angle to that. And that's an angle that has the same terminal side as our angle here. So what that means is that we can find out how much revolutions the negative 1320 degrees is away from the least positive core terminal angle. So let's do that by dividing negative 1320 degrees by 360 degrees. When you do that, you'll find that your answer will be approximately equal to negative 3.67. So if we want to find the least positive core terminal angle, that means we'll have to add the integer that's closest to negative 3.67 and that would have been four. So that means to find that anger, we can add four multiplied by 360 degrees to negative 1320. When we do that, our answer equals 120 degrees. So that means the tangent of negative 1320 degrees would be equal to the tangent of 120 degrees. They would have the same value. So that's what we need to figure out. No, if we were to think about it on our unit circle, let's go back to our unit circle here. And on our unit circle, we don't necessarily have our tangent value that's readily available. OK? Because recall that on the unit circle, right, on the unit circle, our coordinates are in the form of the cosine of theater and the sign of theater where the cosine of theater is the X value and the sign of theater is the Y value. What, what do we know about the tangent, the core sign and the sign? Well, recall that the tangent of an angle equals the ratio of its sign to its core sign. So this tells us then that the tangent of 120 degrees must be equal to the sign of 120 degrees divided by the core sign of 120 degrees. So if we can figure both of those out, then we'll be able to find a tangent of 120 degrees. Now recall on our unit circle to find a sign of 120 degrees, we'd be looking at the Y value and you notice that it would correspond to the value of the sign of 60 degrees. And we're talking now about the sign and we can recall that the sign of degrees, which is the same thing as the sign of 60 degrees would have been the square root of three divided by two. We also know that the core sign of 120 degrees is the negative version of the core sign of 60 degrees. So that means it would be negative or half. So now that we have those values that tells us then that the tangent of 120 degrees would be equal to the sine of 120 degrees, the squared of three divided by two divided by the core sign of 100 and 20 degrees, which is negative or half. We can do some uh moving around here, we can rewrite this as a skit of three divided by two multiplied by negative two, divided by one, we can factor 02 from both terms two goes into time two goes into negative two, negative one times. And no, we can multiply the square root of three multiplied by negative one equals the negative square of three and one multiplied by one equals one. Therefore, that tells us that the tangent of 100 and 20 degrees equals the negative square root of three. When we look back in our answer choices, that would have been answer choice. D thanks a lot for watching everyone. I hope this video helped.

Find the exact value of each expression. See Example 3.tan(-1020°)

Key Concepts

Periodic Properties of Trigonometric Functions

Reference Angles

Tangent Function and Its Values

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