Question 472803
{{{10cos((x+1)/(2)) = 3 }}}
<pre>
Divide both sides by 10

{{{cos((x+1)/(2)) = .3 }}}

Use inverse cosine on a calculator to get
the principle value of 1.266103873

That's the 1st quadrant value.  There is also a
4th quadrant answer of -1.266103873.  We will
have to consider both cases.

FIRST QUADRANT value of {{{(x+1)/2}}}

We can add 2pn and get 1st quadrant coterminal 
angles with that.

So we have 

{{{(x+1)/2=1.266103873 + 2pi*n}}}

Solve for x:

Multiply through by 2

{{{x+1=2.532207346 + 4pi*n}}}

Subtract 1 from both sides

{{{x=1.532207346 + 4pi*n}}}

We set that greater than 10

{{{1.532207346 + 4pi*n>10}}}

Solve for n:

{{{4pi*n>8.467792654}}}

{{{n>8.467792654/(4pi)}}}

{{{n>.673845529}}}

The smallest integer value of n is 1. 
So

{{{x=1.532207346 + 4pi*1}}}

x = 14.09857796

That may be the answer, but we have 
to consider the 4th quardrant value
as well to be sure.

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FOURTH QUADRANT value of {{{(x+1)/2}}}

We can add 2pn and get 4th quadrant coterminal 
angles with that.

So we have 

{{{(x+1)/2=-1.266103873 + 2pi*n}}}

Solve for x:

Multiply through by 2

{{{x+1=-2.532207346 + 4pi*n}}}

Subtract 1 from both sides

{{{x=-3.532207346 + 4pi*n}}}

We set that greater than 10

{{{-3.532207346 + 4pi*n>10}}}

Solve for n:

{{{4pi*n>13.53220735}}}

{{{n>1.076858845}}}

The smallest integer value of n that satisfies
that is 2.  So

{{{x=-1.532207346 + 4pi*2}}}

x = 21.60053388
 
------------------

That is larger so the smallest solution for x
that satisfies

{{{10cos((x+1)/(2)) = 3 }}}

and is greater than 10 is 14.09857796.

Edwin</pre>