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 sohcahtoa the sole part in sohcahtoa is associated with sine sine of the angle Theta is equal to the opposite side divided by the hypotenuse the cop part in sohcahtoa is cosine cosine Theta is adjacent over hypotenuse and tangent Theta that's the Tall part is opposite over adjacent so those are the three trig ratios you need to be familiar with next we have the reciprocal identities cosecant Theta is 1 over sine Theta and secant Theta is 1 over cosine Theta and tangent Theta is one over cotangent Theta so notice this equation here cosecant is one over sine the reverse is also true sine is one over cosecant here we have secant is one over cosine cosine beta is one over secant Theta and for this one tangent Theta is one over cotangent Theta which means cotangent Theta is also one over tangent Theta so those are the reciprocal identities now let's work on an example problem so let's say we have the three four five right triangle and we want to find the six trigonometric ratios for this triangle we could start with sine Theta sine is going to be opposite over hypotenuse so it's four over five cosine Theta is going to be adjacent over hypotenuse so it's 3 over 5. tangent Theta is opposite over adjacent so it's four over three 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 one over tangent it's 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 four over three now cosecant is the reciprocal of sine so cosecant is one over sine and if sine is four over five cosecant is going to be that fraction flipped so it's going to be five over four secant is the reciprocal of cosine so secant is five over three cotangent being the reciprocal of tangent that's going to be three over four 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 one so that's the first Pythagorean theorem I mean Pythagorean identity rather they 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 one over sine is cosecant so we get this identity one plus cotangent squared is equal to cosecant squared so that's the second by factoring 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 one and we know one over cosine is secant so this becomes secant squared so the other Pythagorean identity is this one 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 is 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 signs 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 equal to negative 10 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 cofunction identities cosine Theta is sine 90 degrees minus Theta you could replace 90 with pi over 2. so remember 180 degrees is equal to Pi if you divide that by two you get 9 degrees is equal to pi over two 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 cofunction of secant is cosecant so cosecant Theta is going to be secant 90 minus Theta the cofunction 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 zero degrees is equal to sine 90. both of these are equal to one cosine of 10 degrees is equal to the 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 of 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 cofunctions of each other they will have the same value now let's move on to the double angle identities sine two 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 a 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 two Theta tangent two Theta is two tangent Theta over one 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 of a 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 one plus cosine so tangent Theta over 2 is just square root 1 minus cosine Theta over 1 plus cosine Theta now tangent Theta over two 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 one 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 by 1 minus cosine and whatever you do to the bottom we 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 will get 1 times 1 which is one and then here we have 1 times negative cosine which is negative cosine and then cosine times one 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 one 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 one 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 of 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 one which is equal to 1 minus 2 sine squared theta all we 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 beginning sine squared plus cosine squared is one 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 squares starting from that equation cosine squared is one 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 one 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 four cosine two Theta you can easily derive the other two forms so you don't have to commit all of them to remember Matrix 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 flips and then one 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 two Theta over 2. so in order to reduce the power from Square to not squared or to the first Power you need to double the angle from Theta to two Theta tan squared theta is 1 minus cosine two Theta over 1 plus cosine Z Theta so remember tan is sine of a cosine so if you divide sine squared over cosine squared all you need to do is cancel the 2 and you get 1 minus cosine two Theta over one plus cosine two Theta so if you know the first two power reduce and 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 one 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 product to 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 poverties and 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 two times cosine Alpha minus beta over 2. and then cosine Alpha minus cosine beta is negative two sine Alpha plus beta over two times sine Alpha minus beta over two 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 INE 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 Mis inside or the 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 lower KC is equal to a squared plus b squared minus 2 a b cosine of angle C that's capital c so if you know all three sides you could find angle C and then once you have at least one angle you could use the Law of Sines to solve all the rest of the triangle now to calculate the area of a triangle using trach it's one-half a b sine of angle C another way you can calculate the area of a triangle is used in huron'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 it'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 it exists and here it is for those of you who are curious a minus B over a plus b is equal to tangent one half Alpha minus beta or capital A minus B over tangent one-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 and trig thanks for watching