Question

What are coordinates of the terminal point determined by T = 20 pi /3

Answer 1

(-1/2, squareroot3/2)

Answer 2

The terminal point determined by an angle is given by the point on the circumference of the unit circle such that the angle between the radius of the circle and the x-axis is the given angle.

Given the angle

`t= (20\pi)/(3)`
This angle is equivalent to the angle

`t= (2\pi)/(3)`

To find the terminal point, we note that the radius of the unit circle at that point makes a right angle with the coordinates of the terminal point.
We also note that angle
`t= (2\pi)/(3)` is on the 2nd quadrant of the coordinate axis. This means that the x-value of the terminal point is negative while the y-value is positive.
We also note that the radius of a unit circle is 1.

To find the x-coordinate of the terminal point, we use the relation

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

Similarly to find the y-coordinate of the terminal point, we use the relation

`\sin\left(\pi- (2\pi)/(3) \right)= (y)/(1) \\ \\ \sin\left( (\pi)/(3) \right)=y \\ \\ y= ( √(3) )/(2)`

Therefore, the coordinates of the terminal point determined by T = 20 pi /3 are

`\left(- (1)/(2) , \, ( √(3) )/(2) \right)`