Question 207936
Given a point (5,-5)on the terminal side of angle <font face = "symbol">q</font> find the value of tan<font face = "symbol">q</font>
<pre><font size = 4 color = "indigo"><b>
Plot the point (5,-5)

{{{drawing(400,400,-7,7,-7,7,

graph(400,400,-7,7,-7,7), 

line(5+.1,-5,5-.1,-5),line(5,-5+.1,5,-5-.1),line(5+.1,-5-.1,5-.1,-5+.1),line(5-.1,-5+.1,5+.1,-5-.1), locate(5,-5,"(5,-5)"))}}}

Draw the radius vector from the origin to
the point (5,-5):

{{{drawing(400,400,-7,7,-7,7,
line(5-.3,-5,5,-5),line(5,-5,5,-5+.3),
graph(400,400,-7,7,-7,7), 
line(0,0,5,-5), locate(5,-5,"(5,-5)"))}}}

Indicate with a curved line the angle <font face = "symbol">q</font> measured 
counter-clockwise from the right side of the 
x-axis around to the radius vector.

{{{drawing(400,400,-7,7,-7,7,
line(5-.3,-5,5,-5),line(5,-5,5,-5+.3),
graph(400,400,-7,7,-7,7), 
arc(0,0,2.5,-2.5,0,315),
line(0,0,5,-5), locate(5,-5,"(5,-5)"))}}}

Draw the x and the y coordinates, forming
a right triangle:

{{{drawing(400,400,-7,7,-7,7, locate(2,.6,"x=5"), locate(5.1,-2.1,"y=-5"),
line(5-.3,-5,5,-5),line(5,-5,5,-5+.3),
graph(400,400,-7,7,-7,7), 
arc(0,0,2.5,-2.5,0,315),
line(0,0,5,-5), locate(5,-5,"(5,-5)"),
triangle(0,0,5,-5,5,0) )}}}

Draw another arc (the red one below) to
represent the REFERENCE angle, the one
with vertex at the origin which is
inside the triangle:

{{{drawing(400,400,-7,7,-7,7,
line(5-.3,-5,5,-5),line(5,-5,5,-5+.3),locate(2,.6,"x=5"), locate(5.1,-2.1,"y=-5"),
graph(400,400,-7,7,-7,7), 
arc(0,0,2.5,-2.5,0,315),
line(0,0,5,-5), locate(5,-5,"(5,-5)"),
triangle(0,0,5,-5,5,0),
red(arc(0,0,3.6,-3.6,315,360))

 )}}}

Since 

{{{TANGENT=(OPPOSITE)/(ADJACENT)}}}

The side OPPOSITE the reference angle (red arc) is 
y=-5 and the ADJACENT side to the reference
angle is x=5, then the tangent of the angle in
standard position which is <font face = "symbol">q</font>,
(black arc), has the tangent {{{y/x}}}, so we 
end up with

tan<font face = "symbol">q</font> = {{{y/x=(-5)/5=-1}}}

Edwin</pre>