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.                                                                         
                                                                                                                                                                                                                                                                                                                                                                                    
 (―8 , 15)

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 X axis and a terminal side passing through the given point, rationalize the denominator when necessary. Our given point is negative 17, 144. And we recall from previous lessons that when we have an ordered pair for a point, we have the X value first and then the Y value. So our X value here is negative 17 and our Y value is 144. So what are the six trigonometric function values of an angle that passes through that point? Well, the six trigonometric function values of an angle we recall from previous lessons that those are going to be the sign of theta, the cosine of theta, the tangent of theta, the cotangent of theta, the secant of theta and the coy can't oops see cos of theta. So those are six trigonometric functions. Now, how do we figure out what their values are? So we recall from those same previous lessons that signed theta is equal to Y divided by R. Well, we know what Y is the cosine of theta is equal to X divided by R. We know what X is, the tangent of theta is equal to Y divided by X. We know both of those cotangent of theta is X divided by YC can of theta is R divided by X and the co C can of theta is R divided by Y. Well, we know X and Y but we don't know R. So we do need to figure out what R is, but we recall from previous lessons that R is equal to the square root of X squared plus Y squared. Hey, we can figure out R. All right. So we're taking the square root of, instead of X, we have negative 17 squared plus instead of Y, we have 144 squared. And we're gonna get some big numbers here. When we take negative 17 and square it, we get 289 and plus when we take 144 and skirt, we get 20, add 289 plus 20,736 and we get 21,025. Taking the square rut of that, we get 145. So R is equal to 145. Fantastic. Now we just need to plug these things in. So for Y divided by RY is 144 divide that by 145 and the sign of theta is 144 divided by 145. Now we've got for the cosine of theta X in the numerator that's negative 17 divided by R in the denominator is 145. We know that the cosine of theta is equal to negative 17 divided by 145 for the tangent of theta that's equal to Y in the numerator, 144 divided by X in the denominator. That's negative 17. We'll put the 17 in the denominator and the negative in the numerator. So the tangent of theta is negative 144 divided by 17. The cotangent of theta X in the numerator negative 17 divided by Y which is 144. So the cotangent of theta equals negative 17 divided by 144. The C can of theta, we have R in the numerator which is 145 divided by X which is negative 17. So we'll put 17 in the denominator and that negative in the numerator. So the C can of theta equals negative 145 divided by 17. And the last one, the co C can of theta, R in the numerator is 145 divided by Y in the denominator is 144. And so the of theta is 144 divided by no 145 divided by 144. And there are our six trigonometric function values of the angle that passes through the point negative 17, 144. Well done. We'll catch you on the next one.

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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. (―8 , 15)