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

a. sin 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:

a. sin 2θ

cot θ = 2, θ lies in quadrant III.

Accepted Answer

Hey, everyone in this problem, we're asked to determine the exact value of the expression using the provided data. We're asked to evaluate sine of two beta. We're told that cotangent of beta is equal to four and that beta lies in quadrant three for given four answer choices. Option A eight divided by 17. Option B 17, divided by eight. Option C the square root of 17 divided by 17 and option D negative eight multiplied by the square root of 17 divided by 17. We're gonna start by writing what we're looking for. So we have the expression sign of two beta, which we want to evaluate what you'll notice is that the angle on this expression is two beta. But the information we're given cotangent of beta, the angle is just beta. What we're gonna do is we're gonna use our double angle identity to write our expression in terms of angles just of beta. So that we can solve this problem and simplify this expression using the given data. So we're called that sign of two beta can be written as two S beta multiplied by cosine beta. So in order to calculate sine of two beta. We need to find sign of beta and cosine of beta. Now recall that sign of beta is going to be equal to Y divided by R A cosine of beta is going to be equal to X divided by R. OK. These are all definitions. So what we need to figure out is the Y value, the X value and these are values OK. So we have three things we're looking for Y R X. Now, what we're told is that co tangent of beta is equal to four. We wanna write this in terms of the X and Y values. Now, we're called that cotangent of beta is equal to one divided by tangent of beta. And so cotangent of beta is going to be X divided by Y. So we know that the X value divided by the Y value is going to be equal to four. Let's think about the other information we're given, right? It can be really easy at this stage to say, OK, X is equal to four, Y is equal to one. But we have to think about what quadrant we're in. If we draw our Cartesian plane, we have quadrant one in the top right, quadrant two in the top left quadrant three in the bottom, left quadrant four in the bottom, right. We're told that beta is in quadrant three. So the bottom left, that means the X values and the Y values are both negative and we're below the X axis and we're to the left of the Y axis. So X and Y are negative if X and Y are both negative, what that means is going back to our cotangent equation. We have that X must be equal to negative four and Y is equal to negative one. We have X and we have Y and you could come up with other X and Y values that give you the exact same ratio. OK? So if X was negative eight and Y was negative two, you would get the same answer of four. What matters is the ratio of the two? OK. So you could take any multiple of these and that's fine. What matters is that ratio between the two? So we're just taking the simplest option. We found our X value, we found our Y value, we also needed an R value. So let's recall how R is related to X and Y. And that's through phare theorem, we can write that R squared is equal to X squared plus Y squared. And you can think about this if you are in your Cartesian plane and you were to draw out a little triangle here with beta, the R value is gonna be the hypotenuse of that triangle and the two sides. Well, it's gonna be the X and the Y values. OK. So that's where it comes from Pythia theorem with the triangle that's specially labeled for these trick functions substituting in our values, we have the R squared is equal to negative four squared plus negative one squared tells us that R squared is equal to 17. We can take the square root, we're gonna get both the positive and the negative route. And it's important when you're working through these problems to remember that you do get both of those routes and then you need to interpret which one makes sense. In this context, R is always going to be positive and we're taking the magnitude. And so we're just gonna take the positive route. So we have that R is equal to the square root of 17. We have our X value, we have our Y value, we have our R value. So now we can write our functions sign and cosign, we have that sign of beta is equal to Y negative one divided by R the square root of and cosine of beta is equal to X negative four divided by R the square root of 17. We found sine of beta, we found cosine of beta and remember why we were looking for those and we were looking for those so that we could write our expression sign of two beta in its exact form. So recall, we wrote that sign of two beta through the double angle, identity is equal to two sin of beta multiplied by cosine of beta. We now have these values, we can substitute them in this is gonna be two multiplied by negative one divided by the square root of multiplied by negative four divided by the squared of 17. And we can simplify this. In the numerator, we have two multiplied by negative one multiplied by negative four. That's gonna give us eight. And in the denominator, the squared of 17 multiplied by the squared of 17 gives us 17. And so the expression we were given sine to beta has an exact value of eight divided by 17. If we compare this to our answer choices, we can see that this corresponds with answer choice. A 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: a. sin 2θ cot θ = 2, θ lies in quadrant III.

Verified Solution

Key Concepts

Cotangent Function

Sine Double Angle Formula

Quadrants and Sign of Trigonometric Functions