Point on a Circle (Using Sine and Cosine)

Point on a Circle (Using Sine and Cosine)

How to write the coordinates of the point on a circle using trigonometric functions (sine and cosine): formula, proof, example, and its solution.

Formula

If r: radius and theta: angle formed by the x-axis and the terminal side, then the coordinates of the point on the circle are (r cos theta, r sin theta).

The coordinates of the point on a circle
are (r cos θ, r sin θ).

r: Radius
θ: Angle formed by the x-axis the terminal side

Proof

Point on a Circle (Using Sine and Cosine): Proof of the Formula - Quadrant I

Set the coordinates of the point as (x, y).
And draw a right triangle like this.

Cosine: CAH.

So cos θ = x/r.

Sine: SOH.

So sin θ = y/r.

Multiply r on both sides.

Then r cos θ = x
and r sin θ = y.

So (x, y) = (r cos θ, r sin θ).

Point on a Circle (Using Sine and Cosine): Proof of the Formula - Quadrant II

If (x, y) is in quadrant II,
draw a right triangle like this.

Then the brown angle
is the reference angle of θ.

So cos θ = cos (brown)
= x/r.

And sin θ = sin (brown)
= y/r.

Multiply r on both sides.

Then r cos θ = x
and r sin θ = y.

So (x, y) = (r cos θ, r sin θ).

Point on a Circle (Using Sine and Cosine): Proof of the Formula - Quadrant III

If (x, y) is in quadrant III,
draw a right triangle like this.

Then the brown angle
is the reference angle of θ.

So cos θ = cos (brown)
= x/r.

And sin θ = sin (brown)
= y/r.

Multiply r on both sides.

Then r cos θ = x
and r sin θ = y.

So (x, y) = (r cos θ, r sin θ).

Point on a Circle (Using Sine and Cosine): Proof of the Formula - Quadrant IV

If (x, y) is in quadrant IV,
draw a right triangle like this.

Then the brown angle
is the reference angle of θ.

So cos θ = cos (brown)
= x / r.

And sin θ = sin (brown)
= y / r.

Multiply r on both sides.

Then r cos θ = x
and r sin θ = y.

So (x, y) = (r cos θ, r sin θ).

Point on a Unit Circle

The coordinates of the point on a unit circle are (cos theta, sin theta).

The unit circle is a circle
whose radius is 1.
(and whose center is the origin.)

So the coordinates of the point on the unit circle are
(cos θ, sin θ).

This means y = cos x shows
how the x value of the point on the unit circle changes
as the central angle changes.

Graphing cosine functions

And y = sin x shows
how the y value of the point on the unit circle changes
as the central angle changes.

Graphing sine functions

Example

Find the coordinates of point P. OP = 8, theta = 150 degrees

Think of point P
as the point on the circle
whose radius is 8.

Then P(8 cos 150º, 8 sin 150º).

To find cos 150º and sin 150º,
draw a right triangle on the coordinate plane
whose reference angle is, 180 - 150, 30º.

So the sides of the right triangle are 1, -√3, and 2.

30-60-90 triangle

Cosine: CAH.
So cos 150º = -√3/2.

Sine: SOH.
So sin 150º = 1/2.

So P(8 cos 150º, 8 sin 150º)
= (8⋅[-√3/2], 8⋅1/2).