Find exact values of the six trigonometric functions of each angl...

Question

Find exact values of the six trigonometric functions of each angle. Rationalize denominators when applicable. See Examples 2, 3, and 5.

-2205°

Accepted Answer

Welcome back. Everyone. In this problem, we want to list the six trigonometric function values of the anger negative 2295 degrees. We want to provide exact values and when necessary rationalize the denominator. No, we're trying to find the six trigonometric function values for the angle stated. And if you recall that means we're looking for its sign its cosine, it's tangent and the reciprocal values to those that is the core C, the C can and the core tangent. But the problem is this angle is not between 0 to degrees. So it's going to prove a bit of a challenge to find the exact value to do that. We will need to find its quote a core terminal angle for negative 2295 degrees. What that means is that if I were to draw a sketch off the unit circle here for that angle negative degrees, we need to find the same angle between 0 to degrees that shares the same terminal side. And the best way to do that is to divide our angle by degrees and then find out how many revolutions we need to get the least positive core terminal angle. So let's go ahead and do that. So negative 2295 degrees divided by 360 degrees equals negative 6.375. That means to find the least positive core terminal angle, we can add seven revolutions to negative 2295 degrees. So that means then that the least positive core terminal anger for negative 2295 degrees equals negative 2295 degrees plus seven multiplied by 360 degrees. No seven multiplied by 360 equals 2520. And the negative 2295 plus 2520 equals degrees. Therefore, any or any trigonometric value we're finding for a negative 2295 degrees will be the same value as 225 degrees. So that's the angle that we're working with. So if I were to put that on our unit circle, let me adjust it for that change. So that means no 2295 degrees would have been in the third quadrant. So it would be somewhere down here. OK. And we're looking forward to 225 degree angle 225 degrees. Now, what do we know about the unit circle? That can help us. Well, again, recall that on the unit circle, on the unit circle, each coordinate is in the form of cosine theater, sine theater, where cosine theater is the X value and sine theater is the Y value. So if we can find what those values are, then we should be able to figure out the rest of values for that angle. Now, since let me write that in red, since the cosine of degrees, if you recall from the unit circle equals a negative square root of two divided by two, and the sign of 225 degrees also equals the negative spirit of two divided by two. That means or that would imply then that the core sign of negative 2295 degrees would also equal the negative square root of two divided by two. And the sign of that angle negative 2295 degrees equals a negative square of two divided by two. Now, we can use the cosine and sine to generate the rest of our trigonometric values. We can do that because if you recall the tangent of an angle equals the sign of that angle divided by the core sign of that angle. So that tells us then that the tangent therefore, the tangent of negative degrees would be equal to the sign of theater negative square of two divided by two divided by the cosine of theater which again is negative spirit of two, divided by two. No, both of those terms are the same. And when we divide something by itself, it equals one. So that would be the tangent of negative 2295 degrees. No, we need to find the reciprocal values. Recall, let me do I have any more space here. I think so. Yes, I do recall that the core of an angle is the reciprocal of its sign. OK. So that tells us then that the cores of negative 2295 degrees would be the reciprocal of sine which was negative the negative square of two divided by two. So that means it would be equal to negative two divided by the square of two. We can simplify it by rationalizing that is multiplying both the numerator and the denominator by the square root of two and negative two multiplied by the square of two equals negative two multiplied by the spirit of two. And in our denominator, the square of two multiplied by the square of two equals two. Both numerator and denominator have a common factor of two with factor two from negative two, that's negative one and two into two goes one time negative one multiplied by the square of two equals the negative square root of two. So that would be the core sequence of the negative 2295 degree. Let me come over to the left. So we can find the value of the C and the core tangent. No recall that the second is the reciprocal of the cosine. We already know that the cosine was also equal to the negative square root of two divided by two. That means it will have the same value as the core C count since the sine had the same value as the cosine. So this tells us then that the C account of negative 2295 degrees would also be equal to the negative square root of two. Finally recall that the court, the, the tangent is the reciprocal of the core tangent. So since the tangent of negative 2295 degrees equals one, then the core tangent of negative 2295 degrees would be equal to one divided by one, which would still be one. Therefore, for the anger negative 2295 degrees, its core tangent is one. It's is the negative square of two. Its core CK is the negative square root of two. Its tangent is one. Its core sign is a negative spirit of two divided by two and its sign is the same value. Negative spirit of two divided by two. Thanks a lot for watching everyone. I hope this video helped.

Find exact values of the six trigonometric functions of each angle. Rationalize denominators when applicable. See Examples 2, 3, and 5. -2205°

Key Concepts

Trigonometric Functions

Unit Circle

Angle Reduction and Coterminal Angles

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