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.

Equation of a Circle

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.