Question 1084769
<pre><b><font size = 4>
Here's a different approach for finding the two points P(x,y)

Since {{{tangent=opposite/adjacent}}}, draw a right triangle whose
opposite side is the numerator of 3/5, which is 3, and whose adjacent 
side is the denominator of 3/5, which is 5.  Then &#952; will be the 
angle with opposite side 3 and adjacent side 5:

{{{drawing(400,2000/7,-1,6,-1,4,triangle(0,0,5,0,5,3),locate(.9,.4,theta),
red(arc(0,0,3,-3,0,31)),
locate(2.4,0,5), locate(5.1,1.6,3))}}}

Calculate the hypotenuse by the Pythagorean theorem:

{{{c^2=a^2+b^2}}}
{{{c^2=5^2+3^2}}}
{{{c^2=25+9}}}
{{{c^2=34}}}
{{{c=sqrt(34)}}}

{{{drawing(400,2000/7,-1,6,-1,4,triangle(0,0,5,0,5,3),locate(.9,.4,theta),
red(arc(0,0,3,-3,0,31)),
locate(2.4,0,5), locate(5.1,1.6,3),locate(2.4,2,sqrt(34)))}}}

We place the triangle on a graph so that the vertex is at the origin,
and draw a circle with center at the origin.  But this circle is not
the unit circle.

{{{drawing(800,800,-7,7,-7,7,triangle(0,0,5,0,5,3),locate(.9,.4,theta),
red(arc(0,0,3,-3,0,31)),
graph(800,800,-7,7,-7,7),locate(5.1,3.2,"P(5,3)"),circle(0,0,sqrt(34)),
locate(2.4,.34,5), locate(5.1,1.6,3),locate(2.4,2,sqrt(34)))}}}

The unit circle has radius 1, but the above circle has radius &#8730;34, so we
divide everything by &#8730;34, to make the circle become the unit circle,
so we have the first value of P(x,y):

{{{drawing(800,800,-7/sqrt(34),7/sqrt(34),-7/sqrt(34),7/sqrt(34),triangle(0,0,5/sqrt(34),0,5/sqrt(34),3/sqrt(34)),locate(.9/sqrt(34),.4/sqrt(34),theta),
graph(800,800,-7/sqrt(34),7/sqrt(34),-7/sqrt(34),7/sqrt(34)),
red(arc(0,0,3/sqrt(34),-3/sqrt(34),0,31)),
locate(5.1/sqrt(34)+.01,3.2/sqrt(34)+.01,P(matrix(1,3,5/sqrt(34),",",3/sqrt(34)))),




circle(0,0,1),
locate(2.4/sqrt(34),.34/sqrt(34),5/sqrt(34)), locate(5.1/sqrt(34),1.6/sqrt(34),3/sqrt(34)),locate(2.4/sqrt(34),2/sqrt(34),1))}}}

Also, since tangent is positive in QIII, we can also reflect everything
across the origin, and the angle is increased by 180°, and get another
point P(x,y).

{{{drawing(800,800,-7/sqrt(34),7/sqrt(34),-7/sqrt(34),7/sqrt(34),triangle(0,0,-5/sqrt(34),0,-5/sqrt(34),-3/sqrt(34)),locate(-.9/sqrt(34),-.4/sqrt(34)+.35,theta),
graph(800,800,-7/sqrt(34),7/sqrt(34),-7/sqrt(34),7/sqrt(34)),

locate(-1.2,-.53,P(matrix(1,3,-5/sqrt(34),",",-3/sqrt(34)))),

red(arc(0,0,3/sqrt(34),-3/sqrt(34),0,31+180)),


circle(0,0,1),
locate(-2.4/sqrt(34)-.2,-.34/sqrt(34)+.2,-5/sqrt(34)), locate(-5.1/sqrt(34)-.1,-1.6/sqrt(34)+.1,-3/sqrt(34)),locate(-2.4/sqrt(34),-2/sqrt(34)+.07,1))}}}

Edwin</pre>