Question 765023
An acute angle θ is in a right triangle with sin θ = two thirds . What is the value of cot θ?
<pre>
Draw a right triangle.

Since the sine is {{{(opposite)/(hypotenuse)}}} 

Take the numerator of {{{2/3}}}, which is 2 and put that on the 
opposite side of <font face="symbol">q</font>, which is side b.

Take the denominator of {{{2/3}}}, which is 3 and put that on 
the hypotenuse, the longest side with is c.

{{{drawing(400,2400/7,-.5,3,-.5,2.5,locate(.8,1.1,c=3),locate(1.1,0,a),
locate(.4,.25,theta), locate( 2.3,1,b=2),red(arc(0,0,1.4,-1.4,0,42)),
triangle(0,0,sqrt(5),0,sqrt(5),2),rectangle(sqrt(5)-.2,0,sqrt(5),.2) )}}}{{{matrix(15,1,

Use,the,Pythagorean,theorem,to,find,the,adjacent, side,a, c^2=a^2+b^2,3^2=a^2+2^2,9=a^2+4,5=a^2,sqrt(5)=a)}}}{{{drawing(400,2400/7,-.5,3,-.5,2.5,locate(.8,1.1,c=3),locate(1,0,a=sqrt(5)),
locate(.4,.25,theta), locate( 2.3,1,b=2),red(arc(0,0,1.4,-1.4,0,42)),
triangle(0,0,sqrt(5),0,sqrt(5),2),rectangle(sqrt(5)-.2,0,sqrt(5),.2) )}}}

Since the cotangent is {{{(adjacent)/(opposite)}}},

cot(<font face="symbol">q</font>) = {{{a/b}}} = {{{sqrt(5)/2}}}

Edwin</pre>