Give all six trigonometric function values for each angle θ. Rat...

Question

Give all six trigonometric function values for each angle θ.  Rationalize denominators when applicable. See Examples 5–7. 
 
 tan θ = ―15/8 , and θ is in quadrant II .

Accepted Answer

Welcome back. I am so glad you're here. We are asked to enumerate the value of the other five trigonometric functions of A given the following trigonometric function and condition rationalize the denominators if necessary. Our given trigonometric function is the tangent of A which is equal to negative 12 divided by five. And we're told that A is in quadrant two. All right. So first of all, what are those other five trigonometric functions of A? Well, those are going to be the sign of a, the cosine of a, the cotangent of a, the C of A in the koi Kent of A. And how do we get from the tangent of, of A to those other five? Well, we recall from previous lessons that the tangent of an angle equals Y divided by X. So we've got our Y value, our Y value is in the numerator that's 12. It might be negative 12 might be positive 12. I'm just going to write 12 and leave a space before that our X value that's in the denominator, that's five, but it could be a positive five or a negative five. I'm going to leave a space in front of that one. And now let's figure out which one of these is the negative one. So we're told that a is in quadrant two. If we draw a really rough sketch of a coordinate plane, we've got a vertical y axis and a horizontal X axis, they come together at the origin in the middle, we have four quadrants. The first quadrant is up into the right of the origin. The second quadrant is up into the left of the origin. The third quadrant is down into the left of the origin and the fourth quadrant is down into the right of the origin. And so we're told a is in quadrant two. Well, in quadrant two, the X values are negative and the Y values are positive. So looking at our X and Y, the negative one here, that's going to be our X. All right. Well, with these other five trigonometric functions, we know that the sign of an angle is Y divided by R. The cosine of an angle is X divided by R. The cotangent of an angle is X divided by Y. The C can of an angle is R divided by X. And the Cosic can of an angle is R divided by Y. So we know our X is and our Ys, but we do not yet know our R. But we can figure out the R, we know that R is equal to the square root of X squared plus Y squared, we know our X and our Y so we can figure out R so we've got R equals the square rot of, instead of X, we've got negative five squared plus instead of Y, we've got 12 squared, negative five squared is a positive 25 plus squared is 144. So we're taking the square root of that simplifying inside the radical 25 plus 144 is and the square rut of 169 is a positive 13. So our R value is now that we know Xy and R, we can fill all of those in and solve. All right. So for the sign of angle A, that's Y divided by Ry in the numerator is 12 divided by R in the denominator. That's 13 cosine of angle A X in the numerator is negative five divided by R in the denominators. 13 cotangent of angle A X in the numerator that's negative five divided by Y in the denominator. That's 12 C can of angle A, we have R in the numerator that's 13 divided by X, that's negative five. We'll put the five in the denominator and the negative in the numerator because that's how it's conventionally written. And for the equant of angle A R in the numerator, that's 13 divided by Y in the denominator. That's 12, there's our five other trigonometric functions of A, the sign of A is 12 divided by 13. The cosine of A is negative 5/13 the cotangent of A is negative 5/12. The C can of A is negative 13 5th and the CO C can of A is 13 12th. Well done. We'll catch you on the next one.

Give all six trigonometric function values for each angle θ. Rationalize denominators when applicable. See Examples 5–7. tan θ = ―15/8 , and θ is in quadrant II .

Key Concepts

Trigonometric Functions

Quadrants and Angle Signs

Rationalizing Denominators

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