Question 1107579
FOR AN APPROXIMATE VALUE:
You use a calculator or computer to find {{{tan^-1(7/25)}}}
and find that it is approximately
{{{x=0.2730}}} (in radians) or {{{x=12.642^o}}} .
Then, you multiply that times {{{3/2=1.5}}} to get
{{{(3/2)x=0.4095}}} or {{{(3/2)x=23.463^o}}} .
Finally, you find the cotangent of that angle.
If your calculator does not have cotangent,
you can find the approximate value for tangent,
{{{tan(23.463^o)=0.434}}} ,
and then find the approximate value for cotangent as
{{{cot(23.463^o)=1/tan(23.463^o)=1/0.434=appxoximately2.304}}}


FOR AN EXACT VALUE:
Step 1:
Look for a list of trigonometric identities.
(A search brings up Wikipedia, and you find what you need there).
Step 2:
Figure out how to express {{{(3/2)x}}} in a way that those trigonometric identities will help you find the answer.
 
Here is how I would do it:
{{{(3/2)x=(3x)/2}}} or {{{(3/2)x=3(x/2)}}} .
Looking at the trigonometric identity formulas,
it seems that the easiest way is to first find {{{sin(3x)}}} {{{and cos(3x)}}} ,
and then to find {{{cot((3x))/2)}}} say {{{theta=3x}}}
and use {{{cot(theta/2)=(1+cos(theta))/sin(theta)}}}
 
At this point, I would check to make sure I had copied the problem right,
because it would be a much kinder problem if it were {{{tan^-1(7/24)}}} .
In that case, the angle would be as shown below:
{{{drawing(520,200,-0.5,25.5,-2,8,
triangle(0,0,24,0,24,7),rectangle(24,0,23.5,0.5),
locate(12,0,24),locate(24.1,4,7),locate(12,4.5,25),
red(arc(0,0,10,10,-16.26,0)),locate(5.1,1.2,red(x)),
locate(2,6,tan(red(x))=7/24)
)}}} with {{{system(sin(red(x))=7/25,sin(red(x))=24/25)}}} .
 
{{{x=tan^-1(7/25)}}} is defined as the angle {{{x}}} , with {{{-90^o<x,90^o}}} ,
such that {{{tan(x)=7/25}}} .
As the tangent is positive, the angle must be positive.
So, {{{x}}} is a slightly smaller angle than the one in the drawing above,
the hypotenuse of the triangle for that case would be
{{{sqrt(25^2+7^2)=sqrt(625+49)=sqrt(674)}}} ,
and we get ugly expressions for sine and cosine:
{{{sin(x)=7/sqrt(674)}}} and {{{cos(x)=25/sqrt(674)}}} .
Using the trigonometric identities for triple angles, we get
{{{sin(3x)=(3-4sin^2(x))sin(x)=(3-4*(49/674))(7/sqrt(674))=(913/337)(7/sqrt(674))}}}
and
{{{cos(3x)=(4cos^2(x)-3)cos(x)=(4*(625/674)-3)(25/sqrt(674))=(239/337)(25/sqrt(674))}}}
Then, using the half-angle identity for cotangent
{{{cot((3x)/2)}}}{{{"="}}}{{{(1+cos(3x))/sin(3x)}}}{{{"="}}}{{{(1+(239/337)(25/sqrt(674)))/((913/337)(7/sqrt(674)))}}}{{{"="}}}{{{(337sqrt(674)+5975)/6391}}}