Question

Which point on the unit circle corresponds to −4π3?

(3√2,−12)

(-12,3√2)

(12,−3√2)

(-3√2,12)

Answer 1

The point on the unit circle corresponds to −4π3 is (-1/2, 3√/2) .Therefore , (-1/2, 3√/2) is correct .

We are aware that any point's coordinate on the unit circle may be found using

( cos(θ) ,sin(θ) )

Considering that into the formula above:

θ= -
`(4\pi )/(3)`

So, all we have to do is enter the specified angle's value θ = -
`(4\pi )/(3)` into the formula above:

( cos(θ) ,sin(θ) )

= ( cos ( -
`(4\pi )/(3)` ) , sin ( -
`(4\pi )/(3)` ) )

= ( cos ( 2
`\pi` -
`(4\pi )/(3)` ) , sin ( 2
`\pi` -
`(4\pi )/(3)` ) )

= ( cos (
`(2\pi )/(3)` ) , sin (
`(2\pi )/(3)` )

= ( -
`(1)/(2)` ,
`(√(3))/(2)` )

Evaluating the options:

(1/2, −3√/2) is in the fourth quadrant, not the third.

(-3√/2, 1/2) is in the second quadrant, not the third.

(-1/2, 3√/2) has the correct sine but the wrong cosine (should be negative).

(3√/2, −1/2) has both the correct cosine and sine for the third quadrant and the angle of −4π/3.

Therefore, the point (3√/2, −1/2) corresponds to −4π/3 on the unit circle.

Question

Which point on the unit circle corresponds to −4π/3 ?

(1/2, −3√/2)

(-3√/2, 1/2)

(-1/2, 3√/2)

(3√/2, −1/2)