# 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

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

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*).

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*).

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*).

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 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

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).