Trigonometric Identities

Hello everyone, my name is Archie and I like chemistry. Now let's get on with the lesson, shall we?

In today's class, we were introduced to trigonometric identities. Trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables. So in shorter terms, it's basically an equation in which all trigonometric functions on both sides of the equation will equal each other if given the same variable, if it is a true trigonometric identity.

To verify, or prove that a given equation is an identity, you must show that one side of the equation equals the other. To do that, you must transform or change some trigonometric functions into other trigonometric functions. We do this using our base trigonometric identities, and what base trigonometric identities are, are basically known or proven trigonometric identities where it's been proven that this trigonometric function is equal to that form of that trigonometric function. Our base trigonometric identities are:

• cscƟ=1/sinƟ
• secƟ=1/cosƟ
• cotƟ=cosƟ/sinƟ
• tanƟ=sinƟ/cosƟ
• cotƟ=1/tanƟ

The best way to convert these functions is to convert them into simplest forms or in forms of cosƟ or sinƟ. Another way we can transform trigonometric functions into other trigonometric functions is by using the Pythagorean identities. These Pythagorean identities are basically proven identities that were derived from the Pythagorean theorem (a^2+b^2=c^2). The Pythagorean identities are as follows:

• cos^2Ɵ+sin^2Ɵ=1
• 1+tan^2Ɵ=sec^2Ɵ
• cot^2Ɵ+1=csc^2Ɵ
• 1=sec^2Ɵ-tan^2Ɵ
• 1=csc^2Ɵ-cot^2Ɵ

One method we learned to use to solve prove these goes in these steps:

1. State side
2. Do the work (change info)
3. State = other side

So something would go like this:

cosƟtanƟcscƟ=1

1. State side: So we state that the left hand side (LHS) = cosƟtanƟcscƟ
2. Do the work (change info): So here is where we convert the complex trigonometric functions (tan, csc, sec, cot) into forms of cos or sin. Now we know that tanƟ=sinƟ/cosƟ and cscƟ=1/sinƟ, so we change them accordingly. So our current work should look like this:
   
   cosƟ(sinƟ/cosƟ)(1/sinƟ)=LHS
   

   After that, we continue to do the work and cancel what cancels, like this:
   

   cosƟ(sinƟ/cosƟ)(1/sinƟ)=LHS
   cosƟsinƟ(1)/cosƟsinƟ=LHS
   

   The cosƟ numerator cancels with the cosƟ denominator, and the sinƟ numerator cancels with the sinƟ denominator, both leaving 1s in their places.
   1/1=LHS
   1=LHS
   
3. State = other side: Now that we know that LHS=1, we must state it = to the other side. So it'll be 1=1.

That's really all there is to it, thank you for taking time to read this, and remember, the mind is sharper than the sword.