College Algebra & Trigonometry, 2018a

9.1. TRIGONOMETRIC FUNCTIONS OF NON-ACUTE ANGLES 385
In the previous diagram, we see the values for the sine and cosine of the quadrantal
angles:
cos 0 ◦ =1 cos90 ◦ = 0 cos 180 ◦ = −1 cos 270 ◦ =0
sin 0 ◦ =0 sin90 ◦ = 1 sin 180 ◦ = 0 sin 270 ◦ = −1
If we take a radius of length 1 and rotate it counter-clockwise in the coordinate
plane, the x and y coordinates of the point at the tip will correspond to the values
of the cosine and sine of the angle that is created in the rotation. Let’s look at an
example in the second quadrant. If we rotate a line segment of length 1 by 120 ◦ ,
it will terminate in Quadrant II.
√
(−0.5, 3
(cos θ, sin θ)
2 ) θ=120 ◦
y =
√
3
2
1
60 ◦
x = −0.5
In the diagram above we notice several things. The radius of length 1 has been
rotated by 120 ◦ into Quadrant II. If we then drop a perpendicular line from the
endpoint of the radius to the x-axis, we create a triangle in Quadrant II. Notice
that the angle supplementary to 120 ◦ appears in the triangle and this allows us to
find the lengths of the sides of the triangle and hence the values for the x and y
coordinates of the point at the tip of the radius.
Whenever an angle greater than 90 ◦ is created on the coordinate axes, simply drop
a perpendicular to the x-axis. The angle created is the reference angle. The values
of the trigonometric functions of the angle of rotation and the reference angle will
differ only in their sign (+, −). On the next page are examples for Quadrants II,
III, and IV.