Evaluate each expression without using a calculator. ...

Question

Evaluate each expression without using a calculator.

cos (2 arctan (4/3))

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. Our expression is the cosine of two multiplied by the arc tangent of 84 divided by 13. Our answer choices are answer choice. A negative 5792 divided by 9345. Answer choice B negative 6887 divided by 7225. Answer choice C negative 438 divided by 509 and answer choice D 618 divided by 967. All right. So we've got this cosine of two multiplied by something and that something is the arc tangent or the inverse tangent of 84 divided by 13. But we recall from previous lessons that when we're taking an inverse trigonometric function in this case, inverse tangent, what we have there is an angle or in other words, this would be the tangent of what angle we'll say theta equals 84 divided by 13. The arc tangent of 84 divided by 13 is equal to that angle. And we can just plug in for an. Now, we can plug in an angle theta and look at this as the cosine of two theta, two theta, the cosine of two theta, that's a double angle. And we recall from previous lessons, the double angle identity where we have the cosine of two. And from the textbook, we've got two A equals and we'll just stick with cosine, that'd be two cosine squared A minus one where our angle A is equal to theta in this case. So our cosine of two theta is equal to two cosine squared theta minus one. So we just need to figure out what the cosine of theta is. And we're pretty close with that. We know that the tangent of theta equals 84 divided by 13. But we also recall from previous lessons that tangent is equal to Y divided by X. So we can see from 84 divided by 13 equals Y divided by X, we have an X value of 13 and A Y value of 84. Well, we know that the cosine of theta equals X divided by R. We just need to figure out our R. Now a quick aside, some uh we know that tangent can be positive in either the first or the third quadrants. If I draw a really quick sketch of a coordinate plane, we've got a vertical Y axis. A horizontal X axis, they come together at the origin tangent is positive in quadrants one and three. So we could be in quadrant one or three with our tangent of theta equals 84 divided by 13. However, we're really talking about the inverse tangent and the range where our inverse tangent is going to be, we recall from previous lessons and from the tables in the textbook, from negative pi divided by two to positive pi divided by two, which is the first and the fourth quadrants. So we can eliminate those options from um quadrant three. It's not going to be a negative 13 and a negative 84 for our X and Y values. They're both positive. We're in quadrant one because of our inverse tangents range, we were limited to quadrants one and four has to be quadrant one. So we can figure out our R. Now we recall that R equals the square root of X squared plus Y squared. So we've got the square root of X is 13 squared plus Y is 84 squared. Now, if you can do this whole part without the use of a calculator kudos. Wonderful, congratulations. Um That's impressive. Seriously, 13 squared is 169 plus 84 squared is 7056. And if you add 169 plus 7056 that's 7225. And the square rat of 7225 is 85. So R here is 85 which means that the cosine of theta equals X divided by R which is 13 divided by 85. And that means we can plug that in for our cosine of theta. So we can bring down that two multiplied by. Now, cosine of theta would be 13 divided by 85. But note that it's cosine squared theta. So we've got a square hour 13 divided by 85 and then subtract a one. All right. So squaring 13 divided by 85 that's going to be 169 divided by 7225. That needs to be multiplied by two. And then we're subtracting one that is going to be. So if we multiply that by two, that's 338 divided by 7225 minus one. And if we take 338 divided by 7225 minus one, that equals negative 6887 divided by 7225. And that is our expression simplified. We look at our answer choices and this matches with answer choice. B well done. We'll catch you on the next one.

Evaluate each expression without using a calculator.
cos (2 arctan (4/3))

Key Concepts

Trigonometric Functions

Inverse Trigonometric Functions

Double Angle Formulas

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