Cofunctions of Complementary Angles - Video Tutorials & Practice Problems

Cofunction Identities

Hey, everyone and welcome back. So up to this point, we've spent a lot of time talking about trigonometric functions and how they relate to right triangles. Now, what we're going to be learning about in this video is this new concept of coun identities as well as complementary angles. Now, this might sound a bit confusing and like something that's completely different from what we've been learning about. So, but if that's how you're feeling, don't sweat it because what we're going to be learning about in this video is that both of these concepts relate to these trigonometric functions that we've already been learning about throughout these videos. And I think you're going to find that both of these concepts are very straightforward and relate really nicely to this big concept of the right triangle. So without further ado let's get right into things now to understand complementary angles, these are just angles that add to 90 degrees. And one of the most basic examples of complementary angles would be the non right angles in a right triangle because in a right triangle, these non right angles are always complementary. And to give an example of this, let's take a look at a right triangle down here, we know that one of these angles in the right triangle has to be 90 degrees because that's what a right triangle is. And because this angle has to be 90 degrees, all the in a triangle have to add to 180 degrees. So that means these two angles have to add to 90 degrees because 90 plus 90 would give you 100 and 80. So that's why these two non right angles are always going to be complementary. Now, the reason it's important to understand complementary angles is because this ties into something called the complementary angle theorem. And the complementary angle theorem says that co functions of complementary angles are equal. Now to understand what co functions are, you need to know that we also call these coun identities and coun identities describe how two trigonometric functions are related to each other. One of the most basic examples of this is the sine and the cosine. These are examples of co functions because if you were to take the sign of some angle, let's say 53 degrees, you could say that this is equal to the cosine of the complementary angle and the complementary angle would just be 90 degrees minus whatever angle we started with, which in this case is 53 degrees and 90 minus 50 comes out to 37 degrees. And that makes sense because we already said these angles were complementary. So the sign of 53 degrees is equal to the cosine of 37 degrees. So the function of some angle is going to be equal to the coun of its complementary angle. And this works for every coun identity. For example, the coun to see would be cosecant and the coun to tangent would be cotangent notice for each of the co functions, all we do is put the word co in front of the original function like sequent and cosecant or sine and cosine. So that's an easy way. You can remember what the coun is for the original function. So if we go back to this right triangle that we have and we have the sign of 37 degrees, and we want to find what the coun is, we know the coun would be the cosine. And all we need to do is find the complementary angle which would be 90 degrees minus whatever angle we started with. In this case, 37 degrees and 90 minus 37 comes out to 53 degrees. So the sign of 37 degrees is equal to the cosine of 53 degrees. And if you found the ratios for this triangle for the sign of 37 degrees and the cosine of 53 degrees, you'd find both of these come out to 3/5. So that's the main idea behind coun identities and complimentary angles. Now, to make sure we understand this concept let's try some examples. So in these examples, we are asked to write the expression in terms of the appropriate coun. So let's start with example, a where we're asked to find the coun for the sine of 30 degrees. Well, we know that the coun for sine is cosine. And what we need to do is find the cosine of the complementary angle. Well, the complementary angle would just be our original angle subtracted from 90 degrees. So we would have 90 degrees minus our original angle which is 30 degrees and 90 minus 30 comes up to 60 degrees. So the cosine of 60 degrees would be the answer to example. A. Now let's take a look at example, B where we have the tangent of 16 degrees. Well, the coun for tangent is cotangent and then we need to find the complementary angle which would be 90 degrees minus the original angle we had, which is 16 degrees, 90 minus 16 comes out to 74. So we'd have the cotangent of 74 degrees. And this would be the solution for B. Now, let's take a look at example, C where we have the sequent of zero degrees. Well, the coun for sequent is cos and this will be the cosecant of the complementary angle which would be 90 degrees minus our original angle, which is zero degrees. Now, this might seem a little bit strange since we're starting with a zero degree angle, but we can solve this, the exact same way we've solved these other problems. So 90 minus zero comes up to 90. So we would just have the cosecant of 90 degrees. But now let's example, d how can we go about solving this problem? Well, this one's a little bit complicated because notice that we have a coun that we're starting with, we have cosine and then we have this pie in here. So this is going to be a little bit tricky. But we can solve this just like we solve these other problems where we first find the coun for cosine. And as we've discussed up here, sine and cosine are co functions. So what we would need here is the sine which is the coun of cosine now rather than having 90 degrees minus our original angle. We can't do that because since we have pi that means we're dealing with radiance rather than degrees. And the radiant equivalent of 90 degrees is pi over two. So what that means is I need to have pi over two minus the original angle which in this case is five pi over 18. So this is really going to be the coun for this example. Now to solve this example, what I can do is I can actually get like denominators and I can do that by multiplying the top and bottom of this fraction on the left side by nine. And this will leave us with S and we're going to have nine times pi which is nine pi over nine times two, which is 18 minus five pi over 18. And notice when doing this, we get the same denominator of 18 for both of these fractions. Now, what I need to do from here is subtract five from nine and five minus nine is four. So we're going to have four and then times pi because this is nine pi minus five pi and this is going to be all divided by 18. Now, at this point, I can reduce this fraction because four is the same thing as two times two and 18 is the same thing as two times nine. I can cancel one of the twos in this fraction, meaning we're going to end up with the sign of two pi over nine. So sign of two pi over nine is the solution to example D and that is how you can solve these types of problems involving coun identity. So I hope you found this video helpful. Thanks for watching and please let me know if you have any questions.

Write the expression in terms of the appropriate cofunction.

$\cos\left(\frac{19\pi}{45}\right)$

• A. $\cos\left(\frac{7\pi}{90}\right)$
• B. $\sin\left(\frac{4031\pi}{45}\right)$
• C. $\sin\left(\frac{7\pi}{90}\right)$
• D. $\cos\left(\frac{4031\pi}{90}\right)$

Write the expression in terms of the appropriate cofunction.

$\cot\left(25\degree\right)$

• A. $\tan\left(65\degree\right)$
• B. $\tan\left(25\degree\right)$
• C. $\cot\left(65\degree\right)$
• D. $\cot\left(25\degree\right)$

Using Cofunction Identities to Solve Equations

Hey, everyone. So recently we discussed the concept of coun 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 coun identities. Now, it turns out in this course, you're going to see situations where there are equations that you cannot solve unless you know the coun identities and how to manipulate them properly. And we're going to be going over the step 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. Now, when solving trigger equations, what you want to do is rewrite one side of the equation, then set the arguments of the function equal. And you can do this using the coun identities. We learned before this just tells us how certain functions are related to their coun. So let's say, for example, we have this equation where we have the sine of X minus 10 is equal to the cosine of X. Well, if we wanted to solve for X in this equation, it would not be easy to do this 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 coun identities, we are actually able to solve this problem because recall that sine and cosine are co functions. So what I can do is take this cosine here and I can rewrite this as the sign of the complementary angle. So notice how we took this X and we changed it to 90 minus X since getting the complementary angle is just taking your original angle and subtracting it from 90. Now, once you've done that, notice how you get the same operation on both sides of the equal sign. And what this means is that you can take the insides of the function and you can set them equal to each other. So we can set X minus 10 equal to 90 minus X. And now solving for X is very straightforward. So what I'm gonna do here is use the basic algebra that we've learned by taking 10 and adding it on both sides of this equation. This look at the tens to cancel on the left side because we have a negative and a positive 10 there. And then what I can do is add X on both sides of the equation. This look at the X is to cancel on the right side because we have a negative and positive X. So what I'm gonna have is X plus X on the left side, which is two X and then on the right side of the equal sign, I'm gonna have 90 plus 10 which is 100. Now to solve for X, I just need to divide two on both sides of the equation. That'll get the twos to cancel, giving us that X is equal to 100 divided by two, 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 coun identities. Now to really make sure that we are solid on this concept. Let's go ahead and try another more complicated example. And we're going to do this example by the steps. Now, in this example, we're asked to solve for theta in the following equation and we have the cosine of theta is equal to the sine of two, theta minus 30. Now, 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 use coun identities to get the same trig functions on either side of the equals sign. So to do this, we can recall that the sine and cosine are co functions. So what I can do is take this cosine that I see here and set this equal to the s of the complementary angle for theta, which would be 90 degrees minus theta. So we have the sine of 90 minus the is equal to the sine of two theta minus 30. And this would be step one. Now our next step is going to be to set the insides of the function equal because now that we have the same operations assigned on both sides, we can go ahead and take these insides, the functions and set them equal. So we'll have 90 minus the is equal to two theta minus 30. And that's our next step. Now, our last step is going to be solve for the missing variable, which in this case is theta. So what I'm going to do first is add theta on both sides of this equation. This will get the theta to cancel on the left side giving us that 90 is equal to two theta plus theta which is three theta minus 30. Now, what I can do from here is add 30 on both sides of this equation. And this will get the thirties to cancel on the right side giving us 90 plus 30 which is 100 and 20. And that's going to be equal to what we have left over which is three theta. Now, our last step here is going to be to divide this three on both sides of the equation, they'll get the threes to cancel on the right side leaving us with just theta and theta is going to equal 100 20 divided by three, which turns out to be 40. So our angle theta turns out to be 40 degrees. And this is how you can solve for a missing variable or angle using coun identities. So I hope you found this video helpful. Thanks for watching and let me know if you have any questions.

Find the acute angle solution to the following equation involving cofunctions. $\theta$ is in degrees.

$\cos\left(2\theta+15\right)=\sin\left(5\theta+12\right)$

• A. $1\degree$
• B. $4\degree$
• C. $6\degree$
• D. $9\degree$

Find the acute angle solution to the following equation involving cofunctions. P is in degrees.

$\sec\left(\frac{4P}{5}+20\right)=\csc\left(\frac{5P}{8}+\frac{223}{4}\right)$

• A. $5\degree$
• B. $10\degree$
• C. $12\degree$
• D. $15\degree$

Find the acute angle solution to the following equation involving cofunctions. M is in radians.

$\tan\left(\frac{M}{2}+5\right)=\cot\left(M-5\right)$

• A. $\frac{\pi}{2}$
• B. $\frac{\pi}{4}$
• C. $\frac{\pi}{3}$
• D. $\frac{\pi}{6}$