In Exercises 7–14, use the given information to find the exact value of each of the following:

b. cos 2θ

cot θ = 2, θ lies in quadrant III.

Question

In Exercises 7–14, use the given information to find the exact value of each of the following:

b. cos 2θ

cot θ = 2, θ lies in quadrant III.

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 evaluating a cosine of two beta. And we're told that cotangent of beta is equal to four and that beta lies in quadrant three. We have four answer choices. Option A 17 divided by 15, option B 15, divided by 17, option C negative 15, divided by 17 and option D negative 17 divided by 15. We're gonna start by writing the expression we're trying to evaluate which is cosine of two beta. Now, what we're told in the question is information about cotangent of beta. OK. So our expression has a two beta as the angle, whereas the information we have is just a single beta. So what we wanna do is use our double angle identity and write our expression in terms of some functions of just beta. So we have cosine of two beta and recall through your trick identities that we can write this in three different ways. I'm gonna write all three and then we're gonna talk about which one we should use. You can write this as co squared of beta minus sine squared of beta. We could also write this as two cosine squared of beta minus one or as one minus two sine squared beta. OK. So we have three options to choose from the first one, cos squared beta minus sine squared beta. We're actually gonna ignore that one. We're not gonna use that one because in order to use that one, we need to calculate cosine of beta and sin of beta. So that's gonna be a little bit more work. The other two, we only need one value. So in the first one, we only need cosine of beta. And the second one, we only need s sign of beta. So that's gonna save us a little bit of work. So we could choose either one. Um Let's go ahead and use the first one. So the cosine squared beta minus one. So we wanna find cosine of be, let's just write that out. So we remember what we're looking for. We're looking for cosine of beta and we're told that this is gonna be the X value divided by R, OK. So close beta is X divided by R. Now, the information we're given in is about cotangent. So we know cotangent of beta, OK? And recall that tingent of beta is equal to one divided by tangent of beta. And in terms of X Y and R, this is just gonna be equal to X divided by Y. OK. So we know that co change of beta is X divided by Y. And we're told that this is equal to four. Now, this question says that the angle beta is in Q three. So let's go ahead and draw our coordinate plane. Just a little rough sketch. We have Q one in the upper right, Q two in the upper left, Q three in the bottom left and Q four in the bottom. Right. Now, we are told that we are in quadrant three. If we're in quadrant three, we can see that both the X and Y values are going to be negative and we're below the X axis and we're to the left of the Y axis. So, and why are negative? And that's just like you would see off of a regular graph when you're reading a regular graph, those are negative X and Y values. What that means is that if X divided by Y is equal to four and both of them are negative, we can say that X is equal to negative four and Y is equal to negative one. So we have our X value and we needed to find X and we needed to find R, we have our X value. How do I find R? We recall the X Y and R are all related to each other. We're gonna use a formula that's like the Pythia Green theory. So what we know, I recall the R squared is equal to X squared plus Y squared, substituting in our values we have the R squared is equal to negative four squared plus negative one squared. This tells us that R squared is equal to 17 and we can take the square root and write that R is equal to positive or negative the square root of 17. And it's really, really important when you're working through trade problems like this is that you take both the positive and negative square root in different circumstances. Either one could be correct. In this case, we're talking about that value R which is always going to be positive. So we're gonna take the positive route. So we just have R is equal to the square root of 17. So we have our R value, we have our X value. Now we can go ahead and write. So now cosine of beta is going to be equal to X negative four divided by R the square root of 17. We have cosine of beta. We have to remind ourselves what we're actually trying to find. We're trying to find cosine of two beta. We've cho chosen to use the equation two cosine squared of beta minus one. We've found beta. Now it's just a matter of substituting in these values into that expression. So we have again, cosine of two beta is equal to two cosine squared of beta minus one. This is gonna be equal to two multiplied by negative four divided by the squared of 17. Squared minus one. And you can see now all of the trick is done. We've dealt with the identity to convert the double angle into single, single angle. We've figured out what cosine of beta is. All that's left to do here is simplify this expression, which we know how to do. Simplifying. We get two multiplied by 16 divided by 17 minus one. We want a common denominator. So we can simplify our fraction. We can write this as 32 divided by minus. We can write our one as 17 divided by 17. Now, we have this common denominator of 17. We can subtract across the numerator. We have 32 minus 17, which gives us 15 and we have divided by 17. And so the exact value of the expression cosine of two beta that we were given is gonna be 15 divided by 17. If we compare this to our answer choices, we can see that this corresponds with answer choice. B thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 7–14, use the given information to find the exact value of each of the following: b. cos 2θ cot θ = 2, θ lies in quadrant III.

Verified Solution

Key Concepts

Cotangent Function

Trigonometric Identities

Quadrants of the Unit Circle