Question

Find the coordinates of the point (x,y) shown on the unit circle.
X=?

Answer 1

( -
`(1)/(2)`, -
`(√(3) )/(2)` )

Explanation:

x = cos (
`(4\pi )/(3)` ) = - cos (
`(\pi )/(3)` ) = -
`(1)/(2)`

y = sin (
`(4\pi )/(3)` ) = - sin (
`(\pi )/(3)` ) = -
`(√(3) )/(2)`

Answer 2

The coordinates for the point on the unit circle are:

(-1/2, - (√3)/2)

How to get the coordinates of the point?

Remember that for a point on the unit circle that defines an angle θ, the coordinates of the point are given by:

tan(θ) = y/x

Such that:

√(x^2 + y^2) = 1

Then we have two equations to work with.

In this case, θ = (4/3)*pi

Replacing that in the first equation, and solving for y, we get:

tan((4/3)*pi)*x = y

(√3)*x = y

Now we can replace that on the other equation:

√(x^2 + y^2) = 1

√(x^2 + ((√3)*x)^2) = 1

√(x^2 + 3x^2) = 1

√(4*x^2) = 1

±2x = 1

x = ±1/2

Then:

y = (√3)*x = ±(√3)/2

But notice that our point is on the third quadrant, so both of the values for the components must be the negative ones.

Finally, the coordinates are:

(-1/2, - (√3)/2)

If you want to learn more about unit circles, you can read: