Hey, everyone. As you continue to work through problems dealing with your sum and difference identities, you're going to come across a very specific type of problem in which you'll be given trig values and asked to evaluate a sum or a difference based on those. But you won't actually be given the value of any angles. Now this sounds kind of strange and, like, it might be kind of complicated. But you actually already have all of the knowledge and information that you need to solve these types of problems. But it can still be easy to get lost in all of this information. So here, I'm going to break down for you step by step exactly how to solve these sorts of problems and how exactly to interpret all of this information. So let's go ahead and get started. Now looking at our example problem here, we are told to find the sine of a plus b. Given that the cosine of a is equal to 4/5, the sine of b is equal to 5/13, and angle a is in quadrant 4, and angle b is in quadrant 2. Now what do we do with all of that information? Well, let's take a look at our steps here.
Now, in step 1, we are told to expand our identity and then identify any unknown trig values. So let's take a look at what we're doing here. Here, we're trying to find the sine of a+b. Now, this is my sum identity for sine, so I'm going to go ahead and expand that identity out. Now doing that, I end up with sina+b=sina∙cosb+cosa∙sinb. Now, I've done the first part of that step 1, and I've expanded that identity. But now we want to go ahead and identify any of our unknown trig values. Now in my problem, I'm given that the cosine of a is equal to 4/5 and the sine of b is equal to 5/13. So I already know the cosine of a and the sine of b in that expanded identity. So my unknown trig values here are going to be the sine of a and the cosine of b. Those are my unknown trig values. Now that I've completed step number 1, we can move on to step number 2. Now our goal here is to eventually find these missing trig values, so let's continue on in our steps to get there.
Now, in step 2, we are told that from our given info, we want to sketch and label right triangles in the proper quadrant. Now it's important here to pay close attention to the sign of all of our values because they may be different depending on what quadrant our angle is located in. Now here, we're told that angle a is in quadrant 4. So I'm going to go ahead and draw my right triangle and label that angle in quadrant 4, angle a. Now, I'm also told that the cosine of a is equal to 4/5. So using that information, that tells me that my adjacent side is equal to 4 and my hypotenuse is equal to 5. Now both of these values are going to be positive because our hypotenuse will always be positive and this side length here is in the positive x values. Now let's go ahead and label our other triangle. We're told the angle b is in quadrant 2, so I'm going to go ahead and draw another right triangle here in quadrant 2 and label my angle b. Now we're told that the sine of b is equal to 5/13, which tells me that my opposite side is equal to 5 and my hypotenuse is equal to 13. So here we have successfully completed step number 2 having sketched and labeled both of our triangles here.
Now moving on to step number 3, we want to find any missing sides using our Pythagorean theorem. Now looking at my first triangle here, I am missing this one side length. And looking at these values, I actually don't have to use the Pythagorean theorem because I recognize that this is a 3, 4, 5 triangle. So this missing side length is simply 3. But remember, we want to continue paying attention to the sign of our values here. And since this is in the negative y values, this is actually a negative 3. Now let's take a look at our other triangle. We're missing this side length here. Now you may go ahead and use the Pythagorean theorem here, but here I also notice that this is a special right triangle, so this is a 5, 12, 13 triangle. So this missing side length is simply 12. But again, this is a negative 12 because here we're in the negative x values. Now we have completed step number 3. We have found those missing side lengths. And from here, we can solve for unknown trig values that we identified in step number 1.
Now in step 1, we said that our unknown trig values were the sine of a and the cosine of b. So let's go ahead and solve for those. Now for our first triangle, we want to find the sine of this angle a. Now the sine of angle a is going to be that opposite side of negative 3 divided by my hypotenuse of 5. Now in my other triangle here, we want to find the cosine of this angle b. That's my other unknown trig value, the cosine of b. Now the cosine of b is going to be that adjacent side, negative 12, over that hypotenuse 13. So those are my 2 missing trig values. Now from here, we can finally move on to our last step where we can actually plug in all of those values and simplify.
Now here, we want to go ahead and plug in those values that we just found in step number 4, negative threefifths and negative twelve thirteenths. So here, plugging that in, negative 3/5 for the sine of a multiplying that by the cosine of b, negative 12/13. Now I'm adding that together with those trig values that I was already given in my problem statement. The cosine of a, 4/5 that we see up in our problem here, and then the sine of b, which is 5/13 also in that problem statement, 5/13. Now from here it's just algebra and we just need to simplify. Now because I have these fractions, I need to go ahead and multiply them across in order to simplify this further. Now looking at that first fraction, negative 3/5 times negative 12/13, multiplying across gives me a positive 36/65. Then I'm adding that together with my other fraction multiplied across, which is going to give me 20/65. Now I can go ahead and add these two fractions together because they already have a common denominator. So adding these two fractions, I end up with 56/65. And this is my final answer here. The sine of a plus b given all of this information is 56/65. Now I know that these sorts of problems can be really tedious, so make sure that you're taking your time and paying attention to what you're doing in each step. Thanks for watching, and let me know if you have any questions.
