SOLUTION: Find the values of the six trigonometric functions of θ if the terminal side of Ɵ lies on the given line in the specified quadrant y = {{{1/3}}}x, quadrant 3

Let's draw the line y = x




We want to find the six trig functions of the angle θ indicated by
the red arc.





Let's find a convenient point on that line in quadrant 3, the lower left quadrant.

We can avoid a fraction by choosing to find a point where x = -3, so
we substitute -3 for x in

y = x

y = (-3)

y = -1

So a convenient point on that line in quadrant 3 is (-3,-1)



Draw a vertical line from that point up to the x-axis:




That forms a right triangle in quadrant 3.  Let's label the upper
leg of the right triangle the same as the x-coordinate x=-3.
Let's label the green vertical leg of the triangle the same as the
y-coordinate y=-1. We label the hypotenuse of that triangle as r.


  
Next we calculate r by using the Pythagorean theorem:

r² = x² + y²
r² = (-3)² + (-1)²
r² = 9 + 1
r² = 10
r = 

So we label r as 



Then we remember the six trig ratios:

sin(θ) =  =  = 
cos(θ) =  =  = 
tan(θ) =  =  = 
sec(θ) =  =  =  
csc(θ) =  =  = 
cot(θ) =  =  = 3

Edwin