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 = 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:
r = x + y
r = (-3) + (-1)
r = 9 + 1
r = 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