Trigonometry - Unit Circle

What point corresponds to the angle $-\pi$ on the unit circle?

Explanation

The unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system. $-\pi$ is equivalent to $-180^{\circ}$ which corresponds to the point on the unit circle.

What point corresponds to an angle of $\frac{5\pi}{2}$ radians on the unit circle?

Explanation

The unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system. $\frac{5\pi}{2}$ radians is equivalent to $\frac{5\pi}{2}\times \frac{180^{\circ}}{\pi}=450^{\circ}$ . This is a full circle $360^{\circ}$ plus a quarter-turn $90^{\circ}$ more. So, the angle $\frac{5\pi}{2}$ corresponds to the point on the unit circle.

Give the angle between $0^{\circ}$ and $360^{\circ}$ that corresponds to the point .

$180^{\circ}$

$90^{\circ}$

$45^{\circ}$

$120^{\circ}$

$60^{\circ}$

Explanation

The angle $\pi$ or $180^{\circ}$ corresponds to the point .

What point corresponds to the angle $\frac{\pi}{2}$ on the unit circle?

Explanation

The unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system. $\frac{\pi}{2}$ is equivalent to $90^{\circ}$ which corresponds to the point on the unit circle.

Which of the following points is NOT on the unit circle?

$(\frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2})$

$(\frac{-\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

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

Explanation

For a point to be on the unit circle, it has to have a radius of one. Therefore, the sum of the squares of point's coordinates must also equal one.

Let's try the point $(\frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2})$ .

$(\frac{\sqrt{3}}{2})^{2} + (\frac{\sqrt{2}}{2})^{2} = \frac{5}{4} eq 1$

Therefore, this point is not on the unit circle.

When looking at the unit circle, what are the coordinates for an angle of $\frac{\pi}{4}$ ?

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

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

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

$(\sqrt{2},\sqrt{2})$

Explanation

The coordinates of the point on the circle for each angle are $(\cos\theta,\sin\theta)$ .

Since $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$ and $\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$ , the point will be $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$ .

What is the radius of the unit circle?

$\frac{1}{2}$

$\frac{3}{4}$

Explanation

By definition, the radius of the unit circle is 1.

What point corresponds to the angle $\frac{\pi}{3}$ on the unit circle?

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

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

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

Explanation

In order to find the point corresponds to the angle $\frac{\pi}{3}$ on the unit circle we can write:

$t=\frac{\pi}{3}\ or\ 60^{\circ}$

In the unit circle which has the radious of we can write:

$cos\ t=\frac{x}{R}=\frac{x}{1}=x\Rightarrow x=cos(60^{\circ})=\frac{1}{2}$

$sin\ t=\frac{y}{R}=\frac{y}{1}=y\Rightarrow y=sin(60^{\circ})=\frac{\sqrt{3}}{2}$

So the point corresponds to the angle $\frac{\pi}{3}$ on the unit circle is $(\frac{1}{2},\frac{\sqrt{3}}{2})$

What point corresponds to an angle of $540^{\circ}$ on the unit circle?

Explanation

The unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system. $540^{\circ}$ is a full circle $360^{\circ}$ plus a $180^{\circ}$ more. So, the angle $540^{\circ}$ corresponds to the point on the unit circle.