Question

Note: This is NOT a question. This is a tutorial.

How to derive three of the most used trigonometric identities? See comment below to see how!

Answer 1

TRIGONOMETRIC IDENTITIES

One of the most basic (and most used) trigonometric identity is probably \[\sin^2 \theta + \cos^2 \theta = 1\] but have you ever wondered where this came from? It actually came from the Pythagorean Theorem!! Let us draw a right triangle. Let us name the two legs as a and b, then the hypotenuse as c.

Answer 2

The Pythagorean Theorem states that \[a^2 + b^2 = c^2\] which means that the sum of the squares of the legs is equal to the square of the hypotenuse. Now, we will do some \(\mathbf{magic}\). We divide both sides by \(c^2\), or also known as the square of the hypotenuse...that gives us \[\large \begin{align} &\frac{a^2}{c^2} + \frac{b^2}{c^2} = \frac{\cancel{c^2}}{\cancel{c^2}}\\\\ &\frac{a^2}{c^2} + \frac{b^2}{c^2} = 1\\\\ &(\frac{a}{c})^2 + (\frac{b}{c})^2 = 1 \end{align}\] Refer back to the drawing of the triangle. Can you identify what \(\frac{a}{c}\) and \(\frac{b}{c}\) are? These are \(\sin \theta\) and \(\cos \theta\) respectively. So, if we substitute these newly found trigonometric functions, we'll get \[\large \begin{align} (\sin \theta )^2 + (\cos \theta )^2 &= 1\\\\ \sin^2 \theta + \cos^2 \theta &= 1 \end{align}\] This is how to derive one of the most famous trigonometric identites. From this trigonometric identity, we can derive MORE trigonometric identites!

TRIGONOMETRIC IDENTITY #2 \[\large \begin{align} \sin^2 \theta + \cos^2 \theta &= 1\\\\ \frac{\sin^2 \theta }{\sin ^2 \theta} + \frac{\cos^2 \theta}{\sin^2 \theta } &= \frac{1}{\sin^2 \theta}\\\\\\ 1 + \cot^2 \theta &= \csc^2 \theta \end{align}\]
TRIGONOMETRIC IDENTITY #3\[\large \begin{align} \sin^2 \theta + \cos^2 \theta &= 1\\\\ \frac{\sin^2 \theta }{\cos^2 \theta} + \frac{\cos^2 \theta}{\cos^2 \theta } &= \frac{1}{\cos^2 \theta}\\\\\\ \tan^2 \theta + 1 &= \sec^2 \theta \end{align}\]

Answer 3

at trick 2 r u dividing by sin^2theta is there any other way besides diving?

Answer 4

Very good tutorial. I got confused about the thing, but this made me clear :)

Answer 5

you mean trigonometric identity number 2? you can use the same method using the triangle (i mean using a, b and c) but you still have to divide...

Answer 6

but i'll look into finding a method of deriving the formulas without division @REMAINDER

Answer 7

^ I recommend his tutorials to you guys :D

Answer 8

@lgbasallote , good tutorial for beginners ;)

Answer 9

And I thought I was making tutorials for medals 0.o

Answer 10

weren't you? lol

Answer 11

@GOODMAN

Answer 12

YAAAAAAAAAAAAAY, this was exactly what i was looking for!!! THANK YOU THANK YOU THANK YOU!!!!!
btw. @lgbasallote is AWESOME, if you haven't already noticed.

Answer 13

lol :)) thanks ^_^

Answer 14

I finally understand. My tuts are too long. I will get 2*10 medals for two short tuts, but only 1*15 medals for one long tut/

Answer 15

@GOODMAN is just saying crap.

Answer 16

You should start posting more tutorials . They are delectable

Answer 17

i do have more @jsoeung i think it counts to about 15 now...although the 14 are buried deep below my other questions...i had a 2 week break on tutorials because i had to study...so my past tutorials were buried with the questions i asked for studying

Answer 18

but if you have time you can scroll through http://openstudy.com/users/lgbasallote#/users/lgbasallote/asked idk if you've seen them

Answer 19

Oh I see, this is just the first time I've seen one. I 'll be on the lookout from now on! Thanks!

Answer 20

Wheres, @farmdawgnation when @saifoo.khan is saying stuff :S

Answer 21

Nice tutorial, though I've already studied this stuff.

Answer 22

@inkyvoyd Contrary to popular belief, the staff do have lives on the weekend. :P

That said, I do think @saifoo.khan was just joking around with @GOODMAN.