Trigonometric Identity

Ajouté : 2026-05-16

[00:00:00]Hi guys, we have an interesting question on trigonometry on our screen and we are told to prove it. So the question reads, prove that sin 3 theta = 3 sin theta - 4 sin cub theta.

[00:00:24]Where do we even start from? This looks intimidating but trust me we can always have our way around it. In case you've not subscribed to my channel, kindly do so so that you not miss other interesting videos like this. Now let's begin.

[00:00:43]I'm going to begin from the left hand side because with that I can arrive at some important trigonometric identities.

[00:00:51]If you truly want to be excellent at proving trig identities, then you need to memorize all identities and know how to use them. Without knowing these identities, you cannot solve this type of question. Trust me. So, I'm going to say solution.

[00:01:16]So, I'm going to start from the left hand side. I will say proof left hand side to indicate that I'm starting with the left. So I'll say sine 3 theta.

[00:01:34]Okay.

[00:01:36]Now I'm going to rewrite sin 3 theta because I have seen a pattern. Okay. Now let me say it again. If you want to be excellent at proving trigonometric identities then you must be able to do number one to notice patterns. Okay, patterns are very important and secondly you must know how to use trig identities. You must memorize them and you must know how to use them and when to apply them.

[00:02:08]I can write sin 3 theta to be sin 2 theta + theta. Okay. So this is the pattern that is leading me to what we call sum formula for the sine function sum. Okay. So we have sum and difference formulas for s and cosine. But this has led me to the sign formula.

[00:02:39]Okay. Or uh the sum formula for sign.

[00:02:43]Okay. So this is going to be we can write this as or I'm going to say recall whenever we have sin a + b. This can be written as sin a cos b + cos a sin b. Okay. So in this case right now we can use this formula to expand what we have inside this two inside this bracket. Okay. We have two terms inside this bracket and then we can use this formula to expand it. So I can say let a be equal to 2 theta to make it simple and then b can be equal to theta. Okay. So applying that here this not going to be sin 2 theta okay plus theta is not going to be equal to remember we say sin a cos b right so this not going to be sine okay so looking at this the first one is a this not going to be 2 theta the second one is cos b and then our b is the right. So this now becomes cos theta.

[00:04:16]Okay. Then we say plus the next one here is cos a. Okay. So our a is 2 theta as we have it. So this now becomes cos 2 theta. And then we have And then we have sine theta. Okay. Now using having expanded this right now, what next are we going to do? Let me copy it down. So sin 3 theta has been written as all this that we have on the right hand side. We now say sin 2 theta cos theta plus cos 2 theta sin theta. Okay. Now I can see we have what we call double angle formulas here. So sin 2 theta is a double angle formula and cos 2 theta is also a double angle formula. Now what are we going what are we going to do right now? So I'm going to say recall remember that I said earlier for you to be able to prove these identities or to prove trig identities you must memorize some basic identities.

[00:05:35]I'll say recall that sin 2 theta is equal to 2 sin theta cos theta and then cos 2 theta is = 1 - 2 sin² theta okay so I'm going the reason why I am replacing cuz this one here cos 2 theta has about uh three equivalents but I chose the one with a sign because I want to eliminate this cosine. Okay, that's the reason why I specifically chose the one with a sign function.

[00:06:21]So replacing right now sin 2 theta is equal to 2 sin theta cos theta. So I have replaced this one here with it equivalence. You can see wherever you see sin 2 theta it's equal to 2 sin theta cos theta. So I've done the replacement and it's multiplied by cos theta as you can see. Okay. Now plus this cos 2 theta will be replaced by its equivalent which is 1 - 2 sin² theta and then outside we have sin theta.

[00:07:04]So this multiply by cos theta will give us 2 sin theta cos theta. Okay. So cosine * cosine will give us a cosine squ.

[00:07:21]Okay. And then that's it. So we say plus sin theta * 1 is sin theta. And then sin theta * this will give us -2 sin. Okay. So this one is a sin squ.

[00:07:40]This one is a s. Okay. Just like you have this is s to the power of one. So since the bases are the same s and s they are the same sin theta sin theta they are the same you add their powers right this time becomes sin cub theta okay now again we need to eliminate this cosine because in the answer we have only a sign you can see 3 sin theta 4 sin cube there is no cosine in the final solution or in the right hand side which is what we're looking So I'm going to say recall we have a Pythagorean identity that says that the sin^ 2 theta plus cossine^ 2 theta is always= 1. Shifting this sin square to the other side we now have that cossine square theta is equal to 1 - sin^ 2 theta. So with this right now I can replace this cosine^ square theta 1 - sin^ 2 theta and then we can get rid of the cosine. So replacing we have 2 sin theta. So wherever you see cossine^ 2 you put it value which is 1us sin^ 2 theta. And then we have plus sin theta - 2 sin cub theta.

[00:09:15]Expanding what we have in this bracket 2 sin theta * 1 will give us 2 sin theta minus remember this is sin to the power of 1 right you can see this base and that base are just the same. What you do is to add their powers right. So this now becomes 2 sin cub theta.

[00:09:40]Okay 2 sin cub theta and then we have plus sin theta - 2 sin cub theta.

[00:09:54]Okay so we have 2 sin theta plus sin theta. Okay. So we have two sign here and we have one sign. So we have two of this and one they're the same, right?

[00:10:07]This now becomes 3 sin theta and then - 2 sin cub - 2 sin cub can be seen they are like terms. So - 2 - 2 that gives us - 4 sin cub theta. So we started with the left hand side and we have shown the right. We have shown that both sides are equal. So we can say hence proven.

[00:10:39]Okay. So that's it my dear friends. We have proven this identity.

[00:10:45]Okay. See you in the next video where we shall be talking about more trig identities. Thank you and bye.