Point on a Circle: Using Sine and Cosine
How to write the point on a circle by using sine and cosine: 2 formulas, 1 example, and its solution.
Formula
Formula
A point is on a circle.
The radius is r.
And the central angle is θ.
Then the point on the circle is
(r cos θ, r sin θ).
Formula: Unit Circle
A unit circle is a circle
whose radius is 1:
x2 + y2 = 12.
The point on the circle is
(cos θ, sin θ).
Example
Example
Solution
x2 + y2 = 82
So the radius is 8.
Circle: Equation
The radius is 8.
The central angle is 60º.
So point P is
(8 cos 60º, 8 sin 60º).
To find cos 60º and sin 60º,
draw a 30-60-90 triangle
whose sides are 1, √3, 2.
Find cos 60º.
Cosine is CAH:
Cosine,
Adjacent side (1),
Hypotenuse (2).
So cos 60º = 1/2.
So 8 cos 60º = 8⋅[1/2].
Find sin 60º.
Sine is SOH:
Sine,
Opposite side (√3),
Hypotenuse (2).
So cos 60º = √3/2.
So 8 cos 60º = 8⋅[√3/2].
So (8 cos 60º, 8 sin 60º)
= (8⋅[1/2], 8⋅[√3/2]).
8⋅[1/2] = 4
8⋅[√3/2] = 4√3
So (4, 4√3) is the answer.