Simple trigonometry problem

Hello, I am new here but I’m not big on many words so I’m just going to go right to the reason i am here.

I am having a small trigonometry problem.

I have this equation

sin(2x)=0.7cos(2x)

and I am supposed to solve this where x is equal to or between 0 and 2pi

What I have tried is:

0.75*2*cos(x)^2-2sin(x)cos(x)=0

2(0.75cos(x)-sin(x))=0

Now, I see that I have done something wrong because when i tried inputting this into my calculator using the wonderful "Solve" function I get answers nothing like the answers listen in the back of the book. And second, I cannot see any way to get any more than two solutions from this. (Not like the four solutions I am supposed to get)

Now, I am having similar problems with all the other similar problems, so I am missing something important in general here. Anyone

Re: Simple trigonometry problem

How'd you get from .7 to .75?

Divide both sides by cos 2x to get

$\displaystyle \frac{\sin(2x)}{\cos(2x)} = 0.7 \Rightarrow \tan(2x) = 0.7$, then solve.

Re: Simple trigonometry problem

Thanks for your help! The .7 was actually I typo, it was supposed to be .75 though the exact number is not important to my question.

But I believe what I am having trouble with is more basic.

The way I would normally solve this (From the other chapters in my textbook) I would do something like this:

arctan(0.75)/2=x

This then gives me 0.32 and that is indeed one of the answers in my book.

But how can i solve this to give me all four solutions (not using a graph)?

Re: Simple trigonometry problem

Four solutions over what interval? There are, of course, an infinite number of solutions over all real number because tangent is periodic with period $\displaystyle \pi$. On the interval 0 to $\displaystyle 2\pi$ we have both 2x= .65 and 2x= 0.64+ $\displaystyle \pi$ which give x= 0.32 and x= 0.64+ $\displaystyle \pi/2$= 1.89, approximately. Because of the division by 2, if we are looking for x between 0 and $\displaystyle 2\pi$, then we can add $\displaystyle 2x= 0.64+ 2\pi$ so that $\displaystyle x= .32+ \pi= 3.46$ and $\displaystyle 2x= 0.64+ 3\pi$ so that $\displaystyle x= 0.32+ 3\pi/2= 5.03$.

Re: Simple trigonometry problem

Thank you so much for all you're help!

Re: Simple trigonometry problem

That's great, I like your way ti describe answers of this trigonometric problems, in mu opinion these simple trigonometry problems seems so difficult but when we start to digging these problems then they become so simple like cosine law and others.