Converting between trigonometric ratios example: write all ratios in terms of sine

so what do we have over here right all other trig trigonometric ratios in terms of signing I'm gonna begin with the easiest one I'm gonna write sine a in terms of signing oh they meant all other trigonometric ratios okay but I start here and then I okay maybe I'll go to cos so cos of a in terms of sign e now maybe I can do this right cause of a equals sine of 90 minus a yay I'm done for a my what's that this is not really in terms of sine a this is in terms of some other angle called sine of 90 minus a if a had been I don't know three degrees then this will be sine of 60 degrees but if it's some weird angle like I don't know if you know sine 40 that doesn't mean you know sine 50 which is sine of 90 minus a so this is not actually what we want even though this looks short we want to write it in terms of signing not some other angle and this is exactly why they told me I should memorize trigonometric identities so which trigonometric identity connects cos a and sign a the world famous one right which one is that it's cos square a plus sine square a sine square a equals one now that's the identity that connects these two now see that here we have cos square a which means I can find cos square a in terms of sine square a and then I can find cos a let's do that so I'm going to subtract sine square a from both sides so I have cos square a on this side cos square a on this side and that's equal to one minus sine square a subtracting sine squared a on both the sides and now then what is cos a cos is going to be that's right cause it is just root over root over 1 minus sine square a 1 minus sine square a and you can be happy here or you can ask me hey wait a minute if cos square a is equal to 1 minus sine square a then isn't cause a also pasa negative so shouldn't I put a plus or minus sign out over here in other words if I don't know this cos squared a is I don't know four then cos I could be two but it could also be minus two so there is a plus or minus sign over here and you will be correct but given we're in class ten in class ten cos a is basically if you have a triangle like this a right-angled triangle if this angle is a then cos is basically this length by this length right and this length is a positive number this length is a positive number so this length by this length will also be a positive number so but why do I say in class ten this is cos theta or cos a does what we does what what we mean by cause a change when we grow up it kind of does what we're dealing with in class ten is what what I like to call the demo version of trigonometry where cos is this ratio of this side by this side of a triangle which means it will always be positive because this will be positive and this will be positive but later when the full version unlocks of trigonometry you will know that cos they can also take negative values but as of now you don't have to worry about it because it's a ratio of lengths it is a positive value but later cos sine tan all of them can take negative values because the way you think about them will be different so it's update what cause of a is cos of a is root over 1 minus sine square a root over 1 minus sine square a you'll probably use this a lot now let's go to tanni tan a because once I have cos and tan and I already have sine it's easy for me to find the reciprocals because the other three are just reciprocals of these now I've already used this one here I don't need this anymore so it's it's good to get rid of this to make some space bye-bye so turn away so I know already an identity that connects down and sign and that's done a equals sign a by cos a that itself is an identity and I know that I already have a sign a here I have to now find cos it in terms of sine a but I already did that hard work so I can just use so if I use that then I'll get equals sine of ay by root over 1 minus sine squared a sine square a and that's it I'm done because the other three let's draw a nice line over here the other three that's not a nice line a nice line over here the other three are just reciprocals so the easiest of this is actually cosecant a cosecant a because that's just equal to one by signing 1 by signing then we have secant a which is just 1 by cos a and I can just write 1 by this now because I already did that hardwork root of 1 minus sine square a and the same story over here for Tanny the reciprocal of tan is cot of a and that's going to be equal to the reciprocal of this which is root over 1 minus sine square a divided by sine e so notice that whenever you have to find other trigonometric ratios you probably have to find 2 and that's it because the others will be reciprocals even though it seems like you have to find five other ratios you only have to find two