Question

Find the terminal point on the unit circle determined by 2pi/3 radians

Answer 1

Answer:

Terminal point determined by 2π/3 is

$(-\frac{1}{2}, \frac{\sqrt3}{2})$

Step-by-step explanation:

Unit circle can be defined a as circle which has a radius of 1.

We have to find the point on the circle which corresponds to 2π/3.

For ease, lets convert radians into degrees.

2π/3 = 2(180)/3 = 120°

Let make a line AE at 120° from the origin such that it intersects the circle at point E.

Draw a perpendicular from point E to F. such that we get a right angled triangle AFE.

(DIAGRAM ATTACHED BELOW)

We can see that:

<FAE + 120° = 180°

<FAE = θ = 60°

We have to find the base and perpendicular of the formed triangle, which shows that terminal point

Hypotenuse = radius = 1

cos θ = Base/Hypotenuse

(cos 60) = Base/1

Base = 1/2 where x is in negative direction

sin θ = Perpendicular/Hypotenuse

sin 60 = Perpendicular/1

Perpendicular = (√3) /2

So the terminal point is

$(-\frac{1}{2}, \frac{\sqrt3}{2})$

Answer 2

Using the unit circle, it is found that the terminal point for an angle of $\frac{2\pi}{3}$ is given by:

$\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$

The terminal point on the unit circle for an angle of $\alpha$ is given by:

$(x,y) = (\cos{\alpha}, \sin{\alpha}$

$\frac{2\pi}{3}$ is on the second quadrant, as $\frac{\pi}{2} < \frac{2\pi}{3} < \pi$. Hence, the sine is positive and the cosine is negative.

To find the equivalent angle on the first quadrant, we subtract by $\frac{\pi}{2}$, hence:

$\frac{2\pi}{3} - \frac{\pi}{2} = \frac{4\pi - 3\pi}{6} = \frac{\pi}{6}$

Hence, the sine and the cosine are:

$\sin{\left(\frac{2\pi}{3}\rigth)} = \sin{\left(\frac{\pi}{6}\rigth)} = \frac{1}{2}$

$\cos{\left(\frac{2\pi}{3}\rigth)} = -\cos{\left(\frac{\pi}{6}\rigth)} = -\frac{\sqrt{3}}{2}$

Then, the point is:

$\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$

A similar problem is given at https://brainly.com/question/23843479