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

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

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

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

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

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

2

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

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

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

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

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

So
$x = -90^o$

There are lots of other possibilities...

3

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

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

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

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

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

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

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