Evaluating Trigonometric Functions
If we have any angle, θ, in standard position with a point (x, y) on the terminal side of θ
and r=px2+y2>0, then use the following definitions to evaluate the six trigonometric
functions:
sin θ=y
rcos θ=x
r
tan θ=y
x, x 6= 0 cot θ=x
y, y 6= 0
sec θ=r
x, x 6= 0 csc θ=r
y, y 6= 0
The following figure shows us the quadrants and will also help us to evaluate the
functions:
Quadrant II Quadrant I
sin θ: + sin θ: +
cos θ:−cos θ: +
tan θ:−tan θ: +
Quadrant III Quadrant IV
sin θ:−sin θ:−
cos θ:−cos θ: +
tan θ: + tan θ:−
Problem 6.
Determine the value of the six trigonometric functions of the angle, θ.
sin θ=−1
2; cot >0
Solution Step 1:
With values for yand r, we can find the value for x. Remember that ris
a distance and can never be negative. Thus:
