Find the values of the six trigonometric functions for an angle i...

Question

Find the values of the six trigonometric functions for an angle in standard position having each given point on its terminal side. Rationalize denominators when applicable.                                                                         
                                                                                                                                                                                                                                                                                                                                                                                    
 (3 , ―4)

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine the six trigonometric function values of an angle that has its initial side on the positive part of the X axis. And a terminal side passing through the given point, rationalize the denominator when necessary. Our given point is nine negative 40. And we know that when we are given a point, we are given the X value first and then the Y value. So the X value here is a positive nine and the Y value is a negative 40. Now, how about those six trigonometric functions? What are those we recall from previous lessons that those are the sign of theta, the cosine of theta, the tangent of theta, the cotangent of theta in the, of the, and the Kosi can of theta. And we know from those previous lessons that we find the sign of the by taking Y and dividing it by R. We know why the cosine of theta is equal to X divided by R. The tangent of theta is Y divided by X. The cotangent of theta is X divided by Y. The C of theta is R divided by X and the co of theta is R divided by Y. And we know X and Y, we just don't know R. What we do know about R from previous lessons is that it's equal to the square root of X squared plus Y squared. Well, we can solve for R, we're taking the square root of X which is nine squared plus Y which is negative 40 squared. Nine squared is 81 plus negative 40 squared is 1681 plus is 1681. And the square of that is a positive 41. So our R value here is 41. While we know Xy and R, we can plug those in and then we'll get the six trigonometric function values of an angle passing through that point. So for the sign of the, that's equal to Y in the numerator that's negative divided by R which is 41. So the sign of theta equals negative 40 divided by 41. For the cosine of theta, we have X in the numerator that's nine divided by R which is 41. The cosine of theta is equal to nine divided by 41. For the tangent of theta, we have Y in the numerator negative 40 divided by X which is nine and the tangent of theta is equal to negative divided by nine. For the cotangent of theta, we have X in the numerator that's nine divided by Y which is negative 40. We'll put 40 in the denominator and the negative in the numerator. So we've got the cotangent of theta equal to negative nine divided by 40. For the C can of theta, we have R in the numerator which is 41 divided by X which is nine. The C can of theta equals 41 divided by nine. And finally, for the sequent of theta, we have R in the numerator which is 41 divided by Y which is negative 40. We'll put 40 in the denominator negative in the numerator. And the cos of theta is equal to negative 41 divided by 40. And there are the six trigonometric function values of the angle that passes through nine, negative 40. Well done. We'll catch you on the next one.

Find the values of the six trigonometric functions for an angle in standard position having each given point on its terminal side. Rationalize denominators when applicable. (3 , ―4)

Key Concepts

Trigonometric Functions

Distance Formula

Rationalizing Denominators

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