Download Coordinates on the Unit Circle

Coordinates on the Unit Circle
Based upon the unit circle activity, we have found that the
(x, y) coordinates on the unit circle match to the cos and
sin of the angle.
This means that if we know the coordinates of any angle
on the unit circle, we could find sin, cos, and tan of that
angle.
Example: Given that the terminal side of an angle, 𝜃, crosses the
UNIT CIRCLE at the point −
5 1
,
3 3
, what is the sin 𝜃, cos 𝜃, and
tan 𝜃?
Since the point is on the UNIT CIRCLE, the (x,y) matches to (cos 𝜃, sin 𝜃).
sin 𝜃 =
cos 𝜃 =
To find the tan 𝜃, we can use what we know about SOH-CAH-TOA.
Example con’t: Given that the terminal side of an angle, 𝜃, crosses the
UNIT CIRCLE at the point −
5 1
,
3 3
, what is the sin 𝜃, cos 𝜃, and
tan 𝜃?
Another way you can find tan 𝜃 is by a special relationship,
𝐬𝐢𝐧 𝜽
𝐭𝐚𝐧 𝜽 =
.
𝐜𝐨𝐬 𝜽
This means if you divide the value of sin by the value of cos, you will get
the value of tan.
Sin
÷
Cos
keep, change, flip
But what if the point is NOT on the unit circle?
The (x, y) no longer represents the cos and sin of the angle.
We can still use what we observed in the activity to help us…
Any circle can represent the angles of rotation. If a coordinate lies on
the terminal side of any angle, then those values are the horizontal
and vertical sides of a right triangle.
Example: The point (-3, 4) lies on the terminal side of an angle in
standard position.
This means that we have a triangle with a horizontal side of -3
and a vertical side of 4.
Let’s draw this triangle and see if we can find the values of the 3 basic
trigonometric functions.
Let’s try another…
The point (-2, -5) lies on the terminal side of an angle (𝜃).
Find the values of the 3 basic trigonometric functions.