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.

tan [cos⁻¹ (− 1/3)]

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 tangent of the arc cosine of negative 2/5. We are given four answer choices. The first thing I'm going to do is I'm going to sketch a coordinate plane. So my X and Y axes, so that way I know where I'm drawing my angle. I know that's not a perfectly straight line, but that's why it's a sketch from there. I'm gonna take the arc cosine of negative 2/5 and set it equal to an angle. And I'm gonna call that angle alpha because if I can find the value of alpha, I can find the value of the tangent of alpha. And since alpha will equal the arc cosine of negative 2/5 we'll get our answer. So now that we have an equation, I can take the cosine of each side and get the cosine of alpha equals negative 2/5. Since cosine is negative, I know that this angle will lie in quadrant two. So it will be generally here. So from standard position angle, alpha will go like this which will cause a great triangle right about here. So let's see what else we know? I recall that cosine is X divided by R. So if our cosine is negative 2/5 X will equal negative two and R will equal five. So I'm gonna label this on my diagram X. I'm gonna put as a positive two because distance can never be negative. And R which is like our hypotenuse is five. I need to find the value for Y. So I'm gonna use the Pythagorean theorem. So X squared plus Y squared equals R squared, X squared is negative two squared plus Y squared. We don't know equals R squared. So five squared. So four plus Y squared equals 25 subtracting four from both sides, I get Y squared equals 21 taking the square root of both sides Y equals the square root of 21. And this will be positive because in quadrant two, Y is positive. So our Y value is the square root of now that we have found that we can find the tangent of alpha. So recall that tangent is Y divided by X. So our Y value is the squared of 21 our X value is negative two. So the tangent of alpha should be the negative radical 21 divided by two. And that looks like it is answer choice B so that is our answer. Have a nice day.

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

Key Concepts

Inverse Trigonometric Functions

Right Triangle Trigonometry

Trigonometric Identities

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