In this video, we're going to go over some common trig identities that you're going to counter in a typical trigonometry course. So this video is going to be helpful regardless if you're starting trigonometry or if you're studying for your final exam. We're going to go over the formulas that you need to know. So here we have a right triangle. Across the angle theta is the opposite side. This side is adjacent to it. and across the 90 degree angle, or the blue box, is the hypotenuse. So the first set of formulas you need to be familiar with are the three trig formulas associated with this key expression, Soh Cah Toa. The Soh part in Soh Cah Toa is associated with sine. Sine of the angle theta is equal to the opposite side divided by the hypotenuse. The Ka part in SOHCAHTOA is cosine. Cosine theta is adjacent over hypotenuse. And tangent theta, that's the TOA part, is opposite over adjacent. So those are the three trig ratios you need to be familiar with. Next, we have the reciprocal identities. So cosecant theta is 1 over sine theta, and secant theta is 1 over cosine theta, and tangent theta is 1 over cotangent theta. So notice this equation here. Cosecant is 1 over sine. The reverse is also true. Sine is 1 over cosecant. Here we have secant is 1 over cosine, cosine theta is 1 over secant theta. And for this one, tangent theta is 1 over cotangent theta, which means cotangent theta is also 1 over tangent theta. So those are the reciprocal identities. Now let's work on an example problem. So let's say we have the 3, 4, 5 right triangle. And we want to find the six trigonometric ratios for this triangle. We can start with sine theta. Sine is going to be opposite over hypotenuse, so it's 4 over 5. Cosine theta is going to be adjacent over hypotenuse, so it's 3 over 5. tangent theta is opposite over adjacent. So it's 4 over 3. By the way, you need to also be familiar with the quotient identities. And here are the quotient identities. Tangent theta is sine over cosine. and cotangent theta, which is 1 over tangent, is basically the reciprocal of tangent, so it's cosine over sine. So those are the quotient identities. So tangent, if you take sine over cosine, you divide those two, the fives will cancel, you're going to get 4 over 3. Now, cosecant is the reciprocal of sine. So cosecant is 1 over sine. And if sine is 4 over 5, cosecant is going to be that fraction flipped. So it's going to be 5 over 4. Secant is the reciprocal of cosine. So secant is 5 over 3. Cotangent being the reciprocal of tangent, that's going to be 3 over 4. Now the next thing you need to be familiar with are the Pythagorean identities. So think of the Pythagorean theorem. a squared plus b squared is equal to c squared. Here we have sine squared plus cosine squared is equal to 1. So that's the first Pythagorean identity that you need to know. Now if you take that identity And if you divide it by sine squared, you're going to get another identity. Sine squared divided by sine squared is 1. We know that cosine over sine is cotangent, so that's the quotient identity. So cosine squared over sine squared is cotangent squared. 1 over sine is cosecant. So we get this identity, 1 plus cotangent squared is equal to cosecant squared. So that's the second Pythagorean identity you want to be familiar with. Now, instead of dividing all three terms by sine squared, we can also divide it by cosine squared. sine squared over cosine squared, that's going to be tangent squared. Cosine squared over cosine squared is 1, and we know 1 over cosine is secant, so this becomes secant squared. So the other Pythagorean identity is this, 1 plus tangent squared theta is equal to secant squared theta. So those are the three Pythagorean identities that you need to be familiar with. Now the next type of functions we need to go over are the even and odd functions. So there's four odd trigonometric functions, and two of them are even. So if you were to put a negative sign in front of theta, of sine theta, it's going to become negative sine theta. So notice that the outside sign changed from positive to negative. So that makes it an odd function. For an even function, if we were to change theta to negative theta, we're not going to get negative cosine. We get positive cosine. So as you can see, the two sines in front of cosine are identical for an even trigonometric function. So tangent of negative theta is going to be negative tan theta. So here we have positive tan negative theta, and here this is equal to negative tan theta. So that's an odd function. Secant is the reciprocal of cosine. So if cosine is an even function, secant is going to be even as well. The reciprocal of sine is cosecant, so since sine is an odd function, cosecant will be an odd function as well. Now, since tangent is an odd function, the reciprocal of tangent, which is cotangent, that's going to be an odd function as well. cotangent of negative theta is going to be negative cotangent of theta. So those are the even-odd functions. So the two even trigonometric functions are cosine and secant, but the other four are odd. Now let's talk about the co-function identities. Cosine theta is sine 90 degrees minus theta. You can replace 90 with pi over 2. So remember, 180 degrees is equal to pi. If you divide that by 2, you get 90 degrees is equal to pi over 2. Now, in addition to this, the reverse is also true. If cosine is equal to sine 90 minus theta, well, sine theta is equal to cosine 90 minus theta. Here's some other ones you need to know. The co-function of secant is cosecant. So cosecant theta is going to be secant 90 minus theta. The co-function of tangent is cotangent. So cotangent theta is going to be tangent 90 minus theta, which means tangent theta is cotangent 90 minus theta. You can swap them. Now let's talk about how this works and what it means. Whenever the two angles of cosine and sine add up to 90, the two functions will be equivalent. So cosine of 0 degrees is equal to sine 90. Both of these are equal to 1. Cosine of 10 degrees is equal to sine of 80 degrees. 10 plus 80 is 90. Cosine of 20 is equal to sine of 70. cosine of 30 is equal to sine of 60. Both of these are equal to the square root of 3 over 2. Cosine 45 is equal to sine 45. 45 plus 45 is 90, and both are equal to the square root of 2 over 2. cosine 60 is equal to sine 30. Both of them are equal to one half. And cosine 90 is equal to sine zero, both of which are equal to zero. So whenever the two angles add up to 90, the two functions, as long as they're co-functions of each other, they will have the same value. Now, let's move on to the double angle identities. sine 2 theta is equal to 2 sine theta cosine theta. Feel free to write that in your notes. You'll see that on your trigonometric final exam. By the way, for those of you who want practice problems on these questions, check out the links in the description section below. I'm going to put my trig final exam review video in the description section, so feel free to take a look at that when you get a chance. Cosine 2 theta is cosine squared theta minus sine squared theta. This is also equal to 2 cosine squared theta minus 1, and that is also equal to 1 minus 2 sine squared theta. Now the next double angle formula you want to be familiar with is tangent 2 theta. Tangent 2 theta is 2 tangent theta over 1 minus tangent squared theta. Now let's move on to the half angle identities. Sine theta over 2 is plus or minus the square root of 1 minus cosine theta divided by 2. Next, we have cosine theta over 2, and that's going to be plus or minus the square root of 1 plus cosine theta over 2. So these two formulas look very similar. The only difference is with sine theta over 2, you have a minus sign, and with cosine theta over 2, you have a plus sign. So be mindful of that difference. Next, we have tangent theta over 2. Now remember, tangent is sine over cosine. So if you divide these two, the twos will cancel, and you're going to get the square root of 1 minus cosine over 1 plus cosine. So tangent theta over 2 is just square root 1 minus cosine theta over 1 plus cosine theta. Now, tangent theta over 2 has other forms as well. It's also equal to 1 minus cosine theta over sine theta. And it's also equal to sine theta over 1 plus cosine theta. And you could derive those other forms from the first one. So, starting with this form, here's how you can get this one. What you want to do is multiply. by the reciprocal of this denominator. So instead of 1 plus cosine, we're going to multiply it by 1 minus cosine. And whatever you do to the bottom, you must also do to the top. Now, when we multiply these two, because they're the same, it's just going to be 1 minus cosine theta squared. On the bottom, these two, we could FOIL it. We'll get... 1 times 1, which is 1. And then here we have 1 times negative cosine, which is negative cosine. And then cosine times 1, which is positive cosine. Negative cosine plus cosine, they will cancel. And then cosine times negative cosine gives us negative cosine squared. So I'm going to rewrite this like this. Now, if you recall, sine squared plus cosine squared is 1. If you subtract both sides of that by cosine squared, you'll get 1 minus cosine squared is sine squared. Now, we have the square root. The square root of 1 minus cosine squared is just going to be 1 minus cosine. And the square root of sine squared is just going to be sine. So that's how we can get this form of the equation. Now, to get the other form of the equation, instead of multiplying this equation by the square root of 1 minus cosine over 1 minus cosine, change the negative to a positive. So when you multiply these two, just like before, you're going to get 1 minus cosine squared theta. When you multiply those two, you're going to get 1 plus cosine theta, but squared. And we know 1 minus cosine squared is sine squared. and the square root of sine squared is sine, the square root of 1 plus cosine squared is just going to be 1 plus cosine. And that will give us this form of the equation. So that's how you can derive, let me say that again, that's how you can derive the other forms of the half-angle tangent identity. By the way, going back to cosine 2 theta, so we said there were three forms. Cosine squared theta minus sine squared theta, which is equal to 2 cosine squared theta minus 1, which is equal to 1 minus 2 sine squared theta. All you need to know is this form, because if you know it, you can get the other forms. Now remember... the Pythagorean identity that we talked about in the beginning, sine squared plus cosine squared is 1. So sine squared, if we subtract both sides by cosine squared, sine squared is 1 minus cosine squared. Cosine squared, if we subtract both sides by sine squared, starting from that equation, cosine squared is 1 minus sine squared. So starting with this formula, if we replace cosine squared with 1 minus sine squared, these two are additive, and they will give us 1 minus 2 sine squared. So we get this formula. Now, if we replace sine squared with 1 minus cosine squared, starting with this formula, we'll have cosine squared minus 1 minus cosine squared. Distributing the negative sign, we'll have cosine squared minus 1 negative times negative cosine squared. That's going to be positive cosine squared. And then we can add these two, which will give us 2 cosine squared minus 1, giving us this form. So as long as you know this form of the double angle formula for cosine 2 theta, you can easily derive the other two forms. So you don't have to commit all of them to memory, just one. Now the next thing we're going to talk about are the sum and difference identities. So here we go. Sine alpha plus or minus beta is going to be equal to sine alpha cosine beta. plus minus, cosine alpha, sine beta. Now pay attention to the signs. Notice the order. Here the positive sign is on top, and here the positive sign is on top. So that means that when this is positive, this is going to be positive. When this is negative, this is going to be negative. They match. Now let's contrast that to the next formula. Here we have cosine alpha plus or minus beta is equal to cosine alpha cosine beta minus plus sine alpha. sine beta. So when this is positive, this is going to be negative. And when this is negative, this will be positive. So they're opposite to each other. So cosine alpha plus beta is going to be cosine alpha cosine beta minus sine alpha sine beta, for example. Now the next sum and difference identity has to do with tangent. Tangent alpha plus or minus beta is going to be tangent alpha plus or minus tan beta over 1 minus plus tan alpha tan beta. So for instance, let's say if we had a negative sign between alpha and beta. What's the formula going to be? Notice that these two, they match, but it's opposite to what we have here. So tangent alpha minus beta is going to be tangent alpha minus tangent beta. The first two signs match, the other one's going to be flipped, and then 1 plus tangent alpha tangent beta. Next up we have the power reducing formulas. Sine squared theta is 1 minus cosine 2 theta over 2. cosine squared theta is going to be 1 plus cosine 2 theta over 2. So in order to reduce the power from squared to not squared, or to the first power, you need to double the angle from theta to 2 theta. Tan squared theta is 1 minus cosine 2 theta over 1 plus cosine 2 theta. So remember, tan is sine over cosine. So if you divide sine squared over cosine squared... All you need to do is cancel the two and you get 1 minus cosine 2 theta over 1 plus cosine 2 theta. So if you know the first two power reducing formulas, you can divide the third one. Next we have the product to sum formulas. Sine alpha sine beta is one half cosine alpha minus beta minus cosine alpha plus beta. Next we have cosine alpha cosine beta is equal to one-half cosine alpha minus beta plus cosine alpha plus beta. And then it's sine alpha cosine beta, which is one half sine alpha plus beta, and then plus sine alpha minus beta. The last one is cosine alpha sine beta. is 1 half sine alpha plus beta minus sine alpha minus beta. For those of you who want to see examples on how to use these formulas, if you go to the YouTube search bar and type in products of some formulas organic chemistry tutor, you'll see a video that will give you example problems on how to use these formulas. And the same is true for the other formulas. I have videos on half angle identities, double angle identities, sum and difference formulas, or reducing formulas. Just type in the title of the formula and then Organic Chemistry Tutor into the YouTube search bar and it's going to come up if you want to know how to use these formulas. Next up, we have sum to product formulas. sine alpha plus sine beta is 2 sine alpha plus beta over 2 times cosine alpha minus beta over 2. Next, we have sine alpha minus sine beta. and it's going to be 2 sine alpha minus beta over 2, and then cosine alpha plus beta over 2. Next is cosine alpha plus cosine beta, which is 2 cosine alpha plus beta over 2, times cosine alpha minus beta over 2. And then cosine alpha minus cosine beta is negative 2, sine alpha plus beta over 2, times sine alpha minus beta over 2. So those are the sum-to-product formulas. Next, we have Law of Sines. So let's say this is angle A, angle B, angle C. Opposite to angle A is side A. This is side B and side C. The line of angle A over side A is equal to sine of angle B over side B, which is equal to sine of angle C over side C. So that is the law of sines. So if you have a triangle and you want to find a missing side or a missing angle, the law of sines is a good way to solve the triangle. The next formula has to do with the law of cosines. If you have all three sides, but don't have any angles, this formula is useful for finding that angle. C squared, that's lowercase c, is equal to a squared plus b squared minus 2ab cosine of angle C. That's capital C. So if you know all three sides, you can find angle C. And then once you have at least one angle, you can use the law of sines to solve all the rest of the triangle. Now to calculate the area of a triangle using trig, it's 1 half AB sine of angle C. Another way you can calculate the area of a triangle is using Heron's formula. If you type in Heron's formula organic chemistry tutor in the YouTube search bar, you'll see some examples on how to use this. The area is going to be equal to S times S minus A, S minus B times S minus C. And S is basically half of the perimeter of the triangle. That's A plus B plus C divided by 2. So far, we've talked about the law of sines. We've covered the law of cosines. But it turns out that there's something else known as the law of tangents. This formula is not a commonly used formula, but it exists. And here it is for those of you who are curious. a-b over a plus b is equal to tangent. over 1 half alpha minus beta, or capital A minus B, over tangent 1 half A plus B, or alpha plus beta. So as you can see, the law of tangents is more complicated to use compared to the law of sines and the law of cosines. But that's the formula for those of you who are curious. So that's basically it for the trig identities that you need to know for your final exam, or if you're starting a new course in trigonometry. Those are the formulas that you're going to cover throughout a typical course in trig. Thanks for watching.