Cofunctions of Complementary Angles - Online Tutor, Practice Problems & Exam Prep

Understanding cofunction identities and complementary angles is essential in trigonometry. Complementary angles add up to 90 degrees, and cofunctions like sine and cosine are equal for these angles. For example, $\sin \left(53\right)=\cos \left(37\right)$. To solve equations, rewrite one side using cofunction identities, allowing you to set the arguments equal and solve for the variable. This method simplifies complex trigonometric equations effectively.

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 cofunction 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 far, 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 angles 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 180. 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 cofunctions of complementary angles are equal. Now to understand what cofunctions are, you need to know that we also call these cofunction identities, and cofunction identities describe how 2 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 cofunctions because if you were to take the sine 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 53 comes out to 37 degrees, and that makes sense because we already said these angles were complementary. So the sine of 53 degrees is equal to the cosine of 37 degrees. So the function of some angle is going to be equal to the cofunction of its complementary angle. And this works for every cofunction identity. For example, the cofunction to secant would be cosecant, and the cofunction to tangent would be cotangent. Notice for each of the cofunctions, all we do is put the word "co" in front of the original function, like secant and cosecant, or sine and cosine. So that's an easy way you can remember what the cofunction is for the original function. So if we go back to this right triangle that we have and we have the sine of 37 degrees and we want to find what the cofunction is, we know the cofunction 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 sine of 37 degrees is equal to the cosine of 53 degrees. And if you found the ratios for this triangle for the sine of 37 degrees and the cosine of 53 degrees, you'd find both of these come out to 3/5ths. So that's the main idea behind cofunction identities and complementary 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 cofunction. So let's start with example a, where we're asked to find the cofunction for the sine of 30 degrees. Well, we know that the cofunction 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 out 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 cofunction 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 example b. Now let's take a look at example c where we have the secant of 0 degrees. Well, the cofunction for secant is cosecant, and this will be the cosecant of the complementary angle, which would be 90 degrees minus our original angle, which is 0 degrees. Now this might seem a little bit strange since we're starting with a 0 degree angle, but we can solve this the exact same way we've solved these other problems. So 90 minus 0 comes out to 90, so we would just have the cosecant of 90 degrees.

But now let's try example d. How can we go about solving this problem? Well, this one's a little bit complicated because notice that we have a cofunction that we're starting with, we have cosine, and then we have this pi in here. So this is going to be a little bit tricky. But we can solve this just like we've solved these other problems where we first find the cofunction for cosine. And as we've discussed up here, sine and cosine are cofunctions, so what we would need here is the sine, which is the cofunction 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 radians rather than degrees. And the radian equivalent of 90 degrees is pi/2. So what that means is I need to have π2 minus the original angle which in this case is 5π18. So this is really going to be the cofunction for this example. Now to solve this exampl

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 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.

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. $\frac{\pi}{6}$

Here’s what students ask on this topic:

What are cofunction identities in trigonometry?

Cofunction identities in trigonometry describe the relationship between pairs of trigonometric functions whose angles are complementary, meaning they add up to 90 degrees. For example, the sine and cosine functions are cofunctions because $\sin \left(A\right)=\cos (90°-A)$. Similarly, the tangent and cotangent, secant and cosecant are also cofunctions. These identities are useful for solving trigonometric equations and simplifying expressions.

How do you find the cofunction of an angle?

To find the cofunction of an angle, you need to identify the complementary angle, which is $90°-\theta$. For example, if you have $\sin (30°)$, the cofunction is $\cos (60°)$ because $90°-30°=60°$. This method applies to all trigonometric functions and their cofunctions.

Why are cofunction identities important in solving trigonometric equations?

Cofunction identities are important in solving trigonometric equations because they allow you to rewrite one side of an equation in terms of the same trigonometric function. This makes it possible to set the arguments equal and solve for the variable. For example, if you have $\sin \left(x\right)=\cos \left(y\right)$, you can use the identity $\cos \left(y\right)=\sin (90°-y)$ to rewrite the equation and solve for $x$ or $y$.

Can you provide an example of solving a trigonometric equation using cofunction identities?

Sure! Let's solve $\sin (x-10°)=\cos \left(x\right)$. Using the cofunction identity $\cos \left(x\right)=\sin (90°-x)$, we rewrite the equation as $\sin (x-10°)=\sin (90°-x)$. Now, set the arguments equal: $x-10=90-x$. Solving for $x$, we get $2x=100$, so $x=50$.

What is the complementary angle theorem?

The complementary angle theorem states that the cofunctions of complementary angles are equal. In other words, if two angles add up to 90 degrees, the sine of one angle is equal to the cosine of the other, and vice versa. For example, $\sin (30°)=\cos (60°)$ because $30°+60°=90°$. This theorem is fundamental in trigonometry for simplifying and solving equations.