Hey, everyone. So recently, we discussed the concept of cofunction identities, and how certain trigonometric functions are related to each other through their complementary angles. Now what we're going to be doing in this video is learning about how we can solve equations using cofunction identities. In this course, you're going to see situations where there are equations that you cannot solve unless you know the cofunction identities and how to manipulate them properly. We're going to go over the steps in this video to hopefully make this process seem super straightforward so if you come across this, or I should say, when you come across this on a test, exam, or quiz, you'll know how to solve these problems. So, without further ado, let's get right into things.

When solving trig equations, what you want to do is rewrite one side of the equation to set the arguments of the function equal. You can do this using the cofunction identities we learned before. This just tells us how certain functions are related to their cofunction. So let's say, for example, we have this equation where we have the sine of \(x - 10\) equal to the cosine of \(x\). If we wanted to solve for \(x\) in this equation, it would not be easy to do just with what we're given because there's no real inverse operation we can do on both sides since these are two different trigonometric operations. But if we use the cofunction identities, we are actually able to solve this problem because recall that sine and cosine are cofunctions. What I can do is take this cosine here, and rewrite this as the sine of the complementary angle. Notice how we took this \(x\), and changed it to \(90 - x\) since getting the complementary angle is just taking your original angle and subtracting it from \(90\). Once you've done that, notice how you get the same operation on both sides of the equal sign. This means that you can take the insides of the function, and set them equal to each other. So, we can set \(x - 10\) equal to \(90 - x\), and now solving for \(x\) is straightforward. What I'm going to do here is use the basic algebra that we've learned by adding \(10\) on both sides of this equation. This will get the tens to cancel on the left side, because we have a negative and a positive ten. Then, I can add \(x\) on both sides of the equation. This will get the \(x\)'s to cancel on the right side, because we have a negative and positive \(x\). So what I'm going to have is \(x + x\) on the left side, which is \(2x\), and then on the right side of the equal sign, I'm going to have \(100\). Now to solve for \(x\), I just need to divide \(2\) on both sides of the equation. That'll get the twos to cancel, giving us that \(x\) is equal to \(100\) divided by \(2\), which is \(50\). So the solution for \(x\) is \(50\). Notice how we were able to solve for \(x\) by using this step of the cofunction identities.

Now to really make sure that we are solid on this concept, let's go ahead and try another more complicated example. We're going to do this example by the steps. We're asked to solve for \(\theta\) in the following equation: the cosine of \(\theta\) is equal to the sine of \(2\theta - 30\). For this example, there's no real straightforward way to solve this right off the bat because notice we have two different operations, a cosine and a sine. But what we can do as a first step for solving this problem is to use cofunction identities to get the same trig functions on either side of the equal sign. So, we can recall that the sine and cosine are cofunctions. What I can do is take this cosine I see here, and set this equal to the sine of the complementary angle for \(\theta\), which would be \(90 - \theta\). So we have the sine of \(90 - \theta\) is equal to the sine of \(2\theta - 30\). This would be step one. Our next step is going to be to set the insides of the function equal because now that we have the same operations of sine on both sides, we can go ahead and take these insides of the functions and set them equal. So we'll have \(90 - \theta\) is equal to \(2\theta - 30\), and that's our next step. Our last step is going to be to solve for the missing variable, which in this case is \(\theta\). What I'm going to do first is add \(\theta\) on both sides of this equation. This will get the \(\theta\)'s to cancel on the left side, giving us that \(90\) is equal to \(2\theta + \theta\), which is \(3\theta - 30\). Then, I can add \(30\) on both sides of this equation, and this will get the thirties to cancel on the right side, giving us \(120\), and that's going to be equal to what we have left over, which is \(3\theta\). Now our last step here is going to be to divide \(3\) on both sides of the equation. That'll get the threes to cancel on the right side of the equation, leaving us with \(i\theta\) and \(\theta\) is going to equal \(120\) divided by \(3\), which turns out to be \(40\). So our angle \(\theta\) turns out to be \(40\) degrees. This is how you can solve for a missing variable or angle using cofunction identities. I hope you found this video helpful. Thanks for watching, and let me know if you have any questions.