# Stumped again

• Sep 9th 2007, 02:52 PM
Freaky-Person
Stumped again
Didn't even make it past question one... I have all the formulas, but I never learned how they were made... So basically, I can't modify them.

$\displaystyle \sin 3x = \cos 3x$

It looks so simple!! I would love to be able to do this myself, but I don't know what to do with that 3!!
• Sep 9th 2007, 03:03 PM
Jhevon
Quote:

Originally Posted by Freaky-Person
Didn't even make it past question one... I have all the formulas, but I never learned how they were made... So basically, I can't modify them.

$\displaystyle \sin 3x = \cos 3x$

It looks so simple!! I would love to be able to do this myself, but I don't know what to do with that 3!!

$\displaystyle \sin \theta = \cos \theta$ when $\displaystyle \theta = \frac {\pi}{4} + k \pi$ for some constant $\displaystyle k$

so if $\displaystyle \sin 3x = \cos 3x$ it means that $\displaystyle 3x = \frac {\pi}{4} + k \pi$

now continue
• Sep 9th 2007, 03:04 PM
Soroban
Hello, Freaky-Person!

It's easier than you think . . .

Quote:

$\displaystyle \sin 3x \,= \,\cos 3x$

Divide both sides by $\displaystyle \cos3x\!:\;\;\frac{\sin3x}{\cos3x} \:=\:1 \quad\Rightarrow\quad \tan3x \:=\:1$

. . Then: .$\displaystyle 3x \;=\;\frac{\pi}{4} + \pi n$

Therefore: .$\displaystyle x \;=\;\frac{\pi}{12} + \frac{\pi}{3}n$

• Sep 9th 2007, 05:36 PM
Freaky-Person
Quote:

Originally Posted by Soroban
Hello, Freaky-Person!

It's easier than you think . . .

Divide both sides by $\displaystyle \cos3x\!:\;\;\frac{\sin3x}{\cos3x} \:=\:1 \quad\Rightarrow\quad \tan3x \:=\:1$

. . Then: .$\displaystyle 3x \;=\;\frac{\pi}{4} + \pi n$

Therefore: .$\displaystyle x \;=\;\frac{\pi}{12} + \frac{\pi}{3}n$

I understand the $\displaystyle n$, and the $\displaystyle k$ in the previous post, are some constant, but what constant?

What am I supposed to do with all that? Where did all that come from? T.T
• Sep 10th 2007, 03:29 AM
Jhevon
Quote:

Originally Posted by Freaky-Person
I understand the $\displaystyle n$, and the $\displaystyle k$ in the previous post, are some constant, but what constant?

What am I supposed to do with all that? Where did all that come from? T.T

it works for any and all integers. it is just there to show that the answer repeats itself and infinite number of times. so it works if we add 1 pi, or 2 pi or 3pi or ....

how you deal with the constants depends on the original question. if the question asked for $\displaystyle 0 \leq x \leq 2 \pi$, then you should only choose those integers that cause the solution to fall in that interval, and you are generally expected to list all solutions in this interval, since it is not usually too many.