In Exercises 63–82, use a sketch to find the exact value of each ...

Question

In Exercises 63–82, use a sketch to find the exact value of each expression.

cot (csc⁻¹ 8)

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the following trigonometric expression. By sketching, we are given the co tangent of the ARCO can of six. There are four answer choices knowing that I'm solving this by sketching, I'm going to first draw my X and Y axis, then I'm going to rewrite the co C can. I'm sorry, the art Coy can of six and said it equal to alpha. This way I can find the value of alpha and use it to find the code tangent. So if the ac of six equals alpha, this means that the cos of alpha equals six, I recall that sign of alpha will be equal to the reciprocal of the cosecant of alpha. So if the cose can of alpha is six, which is the same as six divided by one. The sign of alpha would be one divided by six. I can use this to start my sketch because sin is positive. We know this should be in the first quadrant or quadrant one. And I recall that sign is the same as Y divided by R. So one it's gonna be how far up this goes So it goes up one and R is like our hypotenuse of a right triangle and that's gonna equal six and alpha is gonna be in here. We do need to find out what the value for X is. So we can use the Pythagorean theorem, the X squared plus Y squared in this case will equal R squared because that is our hypotenuse. So X squared plus one squared has to equal six squared. So X squared plus one equals 36 subtracting one from both sides, X squared equals 35 and take the square root of both sides. And because this is in quadrant one, we know X is positive. So X is gonna be radical from there. I now know enough to be able to find the cotangent of alpha because I know that the contingent of alpha is the ratio of X divided by Y. So from this sketch, I know that X is radical 35 Y is one. So the cotangent of alpha needs to be radical 35. And that is answer choice D have a nice day.

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In Exercises 63–82, use a sketch to find the exact value of each expression. cot (csc⁻¹ 8)