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.

cos [tan⁻¹ (− 2/3)]

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the following trigonometric expression by sketching the expression we are given is the cosine of the arc tangent of negative 6/7. There are four answer choices. Since I know I'm solving this by sketching, I'm going to first draw an X and Y axis. So I have somewhere to sketch my right triangle and my angle from there. I'm gonna take the part of this that is in the square brackets. So the arc tangent of negative 6/ and I'm gonna set it equal to an angle and I'm gonna call that angle alpha. From here. I can take the tangent of both sides and say that the tangent of angle alpha will equal negative 6/7. Since tangent is negative, our only option is that this angle lies in quadrant four. So we know that this angle goes into quadrant four. So it starts in standard position goes all the way around. So this is where alpha is and we're gonna use quadrant four to draw our right triangle. Recall that tangent is the ratio of Y divided by X. I can say that. Why is negative six and the X is seven. So on my diagram Y is gonna be this vertical line. So that's gonna be six and X is our horizontal line which is seven, we will need to find what are equals we can do. So by saying X squared plus Y squared equals R squared. So we're gonna use the Pythagorean theorem X squared will be seven squared plus Y squared. So six negative six squared equals R squared. So 49 plus 36 equals R squared. It will be 85 equals R squared. We take the square root of both sides, an R will equal radical 85. So adding that onto my diagram R or hypotenuse is radical 85. So now that we know these values in relation to the angle alpha, we can find the cosine of alpha because cosine is the X value divided by the R value. So our X value was seven. Our value is the square root of 85. We want to rationalize our denominator. So we're going to multiply both top and bottom by the square root of 85. And this gives us that the cosine of alpha is seven radical 85 divided by 85. And this is our exact value. And that's answer choice. D have a nice day.

In Exercises 63–82, use a sketch to find the exact value of each expression. cos [tan⁻¹ (− 2/3)]

Key Concepts

Inverse Trigonometric Functions

Right Triangle Relationships

Pythagorean Theorem

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