# Thread: solving for trig x

1. ## solving for trig x

1. sin(x) + sqrt2 = - sinx

2. 3 tan(x/2)+3=0

3. solve for cosx ; cos x =1/2

2. Originally Posted by william
1. sin(x) + sqrt2 = - sinx

2. 3 tan(x/2)+3=0

3. solve for cosx ; cos x =1/2
1. isolate sine
sin(x)+sqrt2=-sinx
2sin(x)=-sqrt2
sin(x)=-sqrt2/2

use the inverse sine function

sin(x)=-sqrt2/2
sin^-1(sin(x))=sin^-1(-sqrt2/2)
x=sin^-1(-sqrt2/2)
x=-0.7853

reference angle must be x=0.7853

sine is negative in quadrant 3 & 4

now take it from here

3. ## Solving for x

Hello william

Originally Posted by william
1. sin(x) + sqrt2 = - sinx

2. 3 tan(x/2)+3=0

3. solve for cosx;
Originally Posted by william
cos x =1/2

1

$\displaystyle \sin x$ to both sides, and subtract $\displaystyle \sqrt{2}$ from both sides:

$\displaystyle 2\sin x = - \sqrt{2}$

$\displaystyle \implies \sin x = -\frac{\sqrt{2}}{2}$

Now I'm not sure what sort of answer is needed now. Do you use a calculator to work out a single answer for
$\displaystyle x$, giving your answer in degrees? Or do you want lots of answers in radians, involving things like $\displaystyle \frac{\pi}{4}$ and $\displaystyle 2n\pi$?

I don't know. But in degrees, possible answers include
$\displaystyle -45^o$, $\displaystyle 225^o$, and so on.

2

$\displaystyle 3 \tan \left(\frac{x}{2}\right) + 3 = 0$

$\displaystyle \implies 3 \tan \left(\frac{x}{2}\right)=-3$

$\displaystyle \implies \tan \left(\frac{x}{2}\right)=-1$

Again, you have lots of possibilities now. One answer is that

$\displaystyle \frac{x}{2} = -45^o$

So
$\displaystyle x = -90^o$

There are lots of other possibilities...

3

By now, I'm guessing that you want all the possible values of
$\displaystyle x$ if $\displaystyle \cos x = \frac{1}{2}$

I reckon the easiest way is to look at a sketch graph of
$\displaystyle y = \cos x$, and work them out from that.

The first positive value is
$\displaystyle 60^o$ (or $\displaystyle \frac{\pi}{3}$ radians), but there's another on the other side of the y-axis, where $\displaystyle x = -60^o$ ($\displaystyle -\frac{\pi}{3}$)

The graph repeats every
$\displaystyle 360^o$, or $\displaystyle 2\pi$ radians, so you get more answers by adding $\displaystyle 360^o$ ($\displaystyle 2\pi$ radians) on to these two answers. So in degrees:

$\displaystyle -60^o, 60^o, 300^o, 360^o, 660^o, 780^o, ...$

I'll leave it to you to find these answers in radians, if you need to.

4. Originally Posted by william
1. sin(x) + sqrt2 = - sinx

2. 3 tan(x/2)+3=0

3. solve for cosx ; cos x =1/2
2.
3 tan(x/2)+3=0
tan(x/2)=-1

-pi=-pi<x<pi=pi
tan^-1(tan(x/2))=tan^-1(-1)
x=2tan^-1(-1)=-1.5707963

therefore the exact solutions are x=2tab^-1(-1)plusminus2npi

substitute -1.5707963 to check

$\displaystyle 3tan\frac{x}{2}+3=3\tan\frac{-1.5707963}{2}+3=0$