JAMES SOUSA: Welcome to our lesson on cofunction identities. Before we discuss the identities, though, I would like to do a quick review so that we can better understand why the identities make sense. So first, the sum of the interior angles of any triangle is 180 degrees or pi radians. 

So we can say the sum of the measure of angles A, B, and C is 180 degrees or pi radians. And then, the two acute angles of a right triangle have a sum of 90 degrees or pi over 2 radians, which means the two acute angles of a right triangle are complementary. So again, if we know this sum is equal to 180 degrees, but the measure of angle C is equal to 90 degrees, that means the measure of angle A plus the measure of angle B must equal 90 degrees or pi over 2 radians, and therefore, angle A and angle B are complementary. 

We also need to be familiar with the right triangle definitions of the six trig functions given here. So if you need to, you may want to pause here. Let's go ahead and talk about cofunction identities. 

A function f is a cofunction of a function g if f of A equals g of B when A and B are complementary angles. So function f and g are different functions, but they're equal to each other as long as angle A and angle B are complementary or have a sum of 90 degrees. So most of the time, you'll see cofunction identities given in the form as we see here in blue. 

First, we have the sine of angle A is equal to the cosine of the quantity 90 degrees minus A, and also, cosine A equals sine of the quantity 90 degrees minus A. What this identity is trying to tell us is that these two trig functions are equal as long as the angles are complementary. Angle A and the angle that measures 90 degrees minus A are complementary angles. And that's why I think it's often helpful to express the cofunction identity as sine A equals cosine B if A plus B equals 90 degrees or pi over 2 radians. 

This equation here takes the place of these two equations here. And then, again, instead of using these two equations, we can say, secant A equals cosecant B if A plus B equals 90 degrees or the two angles A and B are complementary. And the same thing for tangent and cotangent. Tangent A equals cotangent B if A plus B equals 90 degrees. 

Notice how these are all expressed in degrees. But since pi over 2 radians is equal to 90 degrees, we can also express these identities in this form here. Let's go back and take a look at why these identities make sense. We take a look at this green right triangle here. 

Let's go ahead and just assume the measure of angle A is 50 degrees. But we should recognize, that angle A and angle B are complementary, meaning they'd have a sum of 90 degrees. Therefore, the measure of angle B must be 40 degrees. 

Now let's compare the sine of angle A or the sine of 50 degrees to the cosine of 40 degrees. Again, notice how these two angles are complementary. Well, the sine of angle A would be the ratio of the lengths of the opposite side to the hypotenuse, or in this case, A over C. 

Well now, if we switch to angle B and find the cosine of angle B, which, again, is 40 degrees, it's going to be the ratio of the length of the adjacent side to the hypotenuse, which, again, notice, is A over C. Notice the adjacent side to angle B is the same as the opposite side of angle A, and therefore, these two trig function values are equal to each other as long as the two angles are complementary or have a sum of 90 degrees. We can also verify the other identities in a similar way. 

Let's go ahead and take a look at our examples. We want to write each function in terms of its cofunction. So we have the sine of 18 degrees. Well, the cofunction identity for sine involves cosine. So the sine of 18 degrees is equal to the cosine of, let's go ahead and call it angle B. 

And again, the main thing to remember here is that these two angles must be complementary or have a sum of 90 degrees. So angle B plus 18 degrees must equal at 90 degrees. So if we subtract 18 degrees on both sides, we can see that angle B must be 72 degrees. Therefore, the sine of 18 degrees is equal to the cosine of 72 degrees. 

Now, notice, we could have used this identity here and just replaced A with 18 degrees, giving us 72 degrees as well. But I prefer to approach it this way to emphasize that these two angles must be complementary or have a sum of 90 degrees. For our second example, we have tangent 65 degrees that's going to be equal to cotangent of-- again, let's call it angle B-- where these two angles are complementary or have a sum of 90 degrees, which means B plus 65 degrees must equal 90 degrees. Subtracting 65 degrees on both sides, we can see that B is going to be equal to 25 degrees. Therefore, tangent 65 degrees must equal cotangent of 25 degrees. 

Next, we have cosecant 84 degrees. The cofunction identity for cosecant involves secant of angle B, where, again, these two angles are complementary or have a sum of 90 degrees. So we can probably tell by inspection that angle B is going to be 6 degrees. So we have cosecant 84 degrees must equal secant 6 degrees. 

Before we take a look at some involving radians, let's verify these on our calculator. Let's first make sure that we're in degree mode. So we'll press the Mode key. Go down to the third row and highlight Degree. Press Enter. Go back to the Home screen, 2nd, Quit. 

We'll type in sine 18 degrees. Enter. We'll compare that to cosine 72 degrees. Enter. And notice how they are equal to each other. And then we have tangent 65 degrees and cotangent 25 degrees. 

Now, remember, there's no cotangent key, but cotangent theta is equal to 1 over tangent theta. So I'm going to type in 1 divided by, and then, in parentheses, tangent 25 degrees. Again, this is equal to cotangent 25 degrees. And we can see they are equal. So I'll go ahead and stop here, but we can always verify these on a calculator. 

Let's take a look at three more examples that involve radians instead of degrees. Again, the cofunction identity involving cosine also involves sine. It's going to be equal to sine of angle B where these two angles are complementary, or in this case, have a sum of pi over 2 radians. So B plus pi over 4 must equal pi over 2. 

So here, we'll actually show some work. We'll go ahead and subtract pi over 4 radians on both sides of the equation. This would be 0. So we have B equals this difference here. But we do have to have a common denominator, so we have to multiply this fraction by 2 over 2. 

So now, we have 2 pi over 4. And here, we have minus 1 pi over 4. Well, 2 pi over 4 minus 1 pi over 4 is going to be equal to 1 pi over 4 or just pi over 4. So we have cosine pi over 4 is equal to sine pi over 4. 

Let's just take a moment and take a look at this on the unit circle. Here's pi over 4 radians or 45 degrees. Notice, on the unit circle, the x and y-coordinates are the same, verifying that when theta is pi over 4, cosine and sine are equal to each other. Next, we have cotangent pi over 3. That's going to be equal to tangent of angle B where, again, these two angles are complementary or have a sum of pi over 2 radians. So B plus pi over 3 must equal pi over 2. 

Subtract pi over 3 on both sides. So we have angle B must equal this difference here. Our common denominator here is going to be 6. So we'll multiply this by 3 over 3 and this by 2 over 2. So now, we have, what, 3 pi over 6 minus 2 pi over 6, which is going to give us 1 pi over 6 or just pi over 6. So we have cotangent pi over 3 equals tangent of pi over 6. 

And then, for our last example, we have secant pi over 6. This cofunction identity is going to involve cosecant of the angle that is complementary with pi over 6. But we can tell from example 2, pi over 6 and pi over 3 are complementary or have a sum of pi over 2 radians. Therefore, this is going to be cosecant pi over 3 radians. 

And again, we can verify these on the graphing calculator. Or, because all of these are nice reference angles on the unit circle, we could verify these with a unit circle as well. So you may want to pause the video here and verify that these identities here do hold true. OK, I hope you found this helpful. Next, we'll take a look at solving equations involving cofunction identities.