Evaluate each expression without using a calculator. ...

Question

Evaluate each expression without using a calculator.

cos (tan⁻¹ (-2))

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the exact value of the expression without using a calculator. We are given the cosine of the inverse tangent of negative five. Our answer choices are answer choice A the square root of 26 divided by 26 answer choice B negative five square root 26 divided by 26 answer choice C five divided by 26 and answer choice D five square root six divided by 26. All right. So we've got the cosign of something and that something is the inverse tangent of negative five. But we recall from previous lessons that inverse tangent that's just the same as an angle. So we can set angle theta equal to the inverse tangent of negative five. And another way to think about that would be saying that the tangent of angle theta equals negative five. Now we can use substitution where we have the inverse tangent of negative five that's equal to theta. So we can just put theta in there. And really what we're trying to find is the cosine of theta. So what do we know about angle theta? We know the tangent of theta equals negative five. We recall from previous lessons that tangent equals Y divided by X. So we know here that negative five equals Y divided by X or in other words, that's going to be a five divided by one and one of those, the five or the one are going to be negative. But which one is it? Well, let's think for a moment about where tangent is equal to a negative value. If we draw a quick rough sketch of a coordinate plane, we have a vertical y axis, a horizontal X axis, those come together at the origin in the middle. We recall from previous lessons that tangent is negative in the second in fourth quadrants. However, we're not just talking about tangent, we're talking about the inverse tangent and inverse tangent has a limited range because it's a 1 to 1 function. And the range of inverse tangent is from negative pi divided by two to positive pi divided by two. Or in other words, from the negative part of the Y axis to the positive part of the Y axis heading counterclockwise or the 4th and 1st quadrants. So we can disregard that tangent is negative in the second quadrant because inverse tangent can't be in the second quadrant. So here we are definitely talking about the fourth quadrant in the fourth quadrant. Why values are negative? So it is a negative five divided by a positive one and that's helpful because we can set our X equal to one, our Y equal to negative five. And why do we need to do that? Because we're trying to find cosine of theta. And we recall from previous lessons that cosine is equal to X divided by R. So all we need to figure out now is our, well, we recall from previous lessons that R equals the square root of X squared plus Y squared. We know X and Y we can plug those in and solve for R. So this is the square root of X is one squared plus Y is negative five squared. So this is the square root of one squared is one plus negative five squared is 25 1 plus 25 is 26. So we have the square root of 26. That's our R value. Now, we can plug that in. So the cosine of theta equals X in the numerator is one divided by R in the denominator is the square root of 26. Well, that's what we're trying to find. Right cosine of theta equals one divided by the square root of 26. However, we do need to rationalize that. So we'll multiply both the numerator and the denominator by the square root of 26. And when we do that, this will simplify to be the square root of 26 divided by the square root of 26 multiplied by itself. And the denominator is just 26. So this simplifies to be the square root of 26 divided by 26. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

Evaluate each expression without using a calculator.
cos (tan⁻¹ (-2))

Key Concepts

Inverse Trigonometric Functions

Right Triangle Relationships

Pythagorean Identity

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