Use the given information to find the exact value of each of t...

Question

Use the given information to find the exact value of each of the following: $tangent 2 theta$

$\cos θ = \frac{24}{25},$

Accepted Answer

Hey, everyone. Welcome back in this problem. We're asked to determine the exact value of the expression using the provided data. The expression we're asked to evaluate is tangent of two beta are told that the cosine of beta is equal to 21 divided by 29. And that beta lies in quadrant four. We have four answer choices. Option A negative 41 divided by 840. Option B 840 divided by 41. Option C negative 840 divided by 41 option D 41 divided by 840. We're gonna start by writing the expression we're trying to evaluate which is tangent of two beta. Now, we have this angle of two beta in our expression. But the information we're given or the data we're given cosine of beta equals 21 divided by 29. That was just the angle of beta instead of two beta. So what we're gonna do is use our double angle identities to write tangent of two beta in terms of some functions where the angle is just beta. OK. So recall that tangent of two beta is equal to 2 beta divided by one minus 10 square beta. So now we've written it in terms of functions with just beta. And you'll notice that in order to calculate tangent of two beta, we need to first figure out tan beta. So that's what we're gonna do. We're gonna find tan beta. Now, let's write out Tam beta and let's write the definitions and recall that 10 of beta is gonna be the Y value divided by the X value. So we wanna take the information we're given and we wanna try to figure out this Y value and this X value. Now we're told cosine of beta is equal to 21 divided by 29 and recall that cosine of beta is going to be the X value divided by the R value. Now, we have to think about what quadrant we're in K to figure out if X should be positive if X should be negative. So let's first look at that quadrant one, drawing a quick Cartesian plane is the top right quadrant two, top left quadrant three, bottom, left quadrant four bottom, right. We're told that beta is in quadrant four. So we're in the bottom, right. So the X value is going to be positive and we have X positive and we have that Y will be negative and we're to the right of the Y axis, but we're below the X X. So we're looking at cosine of beta, we know that, that's equal to X divided by R. And we're told that in this case, it's 21 divided by 29. We can say that X is equal to 21 and R is equal to 29. And you could take any multiple of these, you would get that same ratio and that's fine. We're just gonna choose these ones. The simplest option where we have that correct ratio. So we found our X value and that's part of that tangent of beta that we needed Y divided by X. But we need to find wi stuff. Now, we're called that Y X and R are all related and they're related to the Pythagorean theorem. You can imagine if we have our Cartesian plane and we draw our angle, OK? We have this R value which is the hypotenuse the distance from the origin. We have our X value and our Y value. OK. So this is just a triangle. We're using Pythagorean theorem and it's just been labeled special for these trig functions. So we can write R squared is equal to X squared plus Y squared. Substituting in our values. 29 squared is equal to 21 squared plus Y squared. Simplify 841 is equal to 441 plus Y squared. We're gonna subtract 441 from both sides to isolate our Y squared. We get that Y squared is equal to 400 and if we take the square root, we're gonna have that Y is equal to plus or minus twice. Now, you have to remember when you are working with these problems that when you take the square route, you get both the positive and negative route. Depending on the situation, depending on which quadrant you're in. You could have either answer being correct. You need to take both of them and then interpret based on the problem. Which one makes sense. In this case, we're told that we are in quadrant four. We've already said that that means X is positive and Y is negative. So we're getting a Y value, we need to take that negative route. Y is equal to negative 20. So we have our X value, we have our Y value. We can now go ahead and write tangent and that's why we needed X and Y was to write tan data. So now we can write that tan beta is equal to Y negative 20 divided by X 21. OK. Great. We have Tam Beta. Remember what we're looking for is not tan beta but tangent of two beta. We can calculate that using the double angle formula, two tan beta divided by one minus 10 square beta, gonna give ourselves some more room to work here. And we're gonna just work out this expression tangent of two beta is equal to two, the engine of beta divided by one minus 10 squared beta. We're gonna substitute in the value. We found we have two multiplied by 20 divided by 21 divided by one minus negative 20 divided by 21 squared. Hey, now this looks a little bit messy. But what you'll notice is that we're done with the trick. OK? We've done our double angle formula. We've figured out what Tamba is. And now all that's left to do is evaluate this fraction. We're just dealing with numbers and fractions now and we know how to do that. If we simplify in the numerator, we have negative 40 divided by that's gonna be divided by one minus 400 divided by 441 in the denominator. We wanna find a common denominator to simplify our fraction. So we can write this as negative 40 divided by 21 divided by one can be written as 441 divided by 441. Then we subtract 400 divided by 441 and we get 41 divided by 441. Now, for dividing two fractions recall that we can multiply by the reciprocal. So we can write this as negative 40 divided by multiplied by 441 divided by 41. And what we wanna do here is just go through and simplify as much as we can. 441 is divisible by 21. So we're gonna divide numerator and denominator by 21. OK? The 21 the denominator goes away and in the numerator turns into 21. So we end up with negative 40 multiplied by 21 divided by 41 which we can simplify to negative 840 divided by 41. And this is the exact value of our expression 10 to be that we were looking for. If we compare this with our answer choices, we see that the correct answer is option C negative 840 divided by 41. Thanks everyone for watching. I hope this video helped see you in the next one.

Use the given information to find the exact value of each of the following: $tangent 2 theta$
$\cos θ = \frac{24}{25},$

Key Concepts

Trigonometric Ratios and Quadrants

Double-Angle Identity for Tangent

Finding tan θ from cos θ

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