Unit Circle

Identify the Quadrant of an Angle in Radians

Given an angle of $ \frac{4\pi}{3} $ radians, determine the quadrant in which the terminal side of the angle lies.

First, recall that the unit circle is divided into four quadrants:

1. Quadrant I: $0 < \theta < \frac{\pi}{2}$

2. Quadrant II: $\frac{\pi}{2} < \theta < \pi$

3. Quadrant III: $\pi < \theta < \frac{3\pi}{2}$

4. Quadrant IV: $\frac{3\pi}{2} < \theta < 2\pi$

Here, $ \frac{4\pi}{3} $ radians is greater than $ \pi $ and less than $ \frac{3\pi}{2}$. Hence, it lies in Quadrant III.

Find the Cosine of an Angle on the Unit Circle

To find the cosine of an angle, we use the unit circle. Given that the angle is $\theta = \frac{\pi}{3}$, we need to find $\cos(\frac{\pi}{3})$.

On the unit circle, the coordinates of the point corresponding to the angle $\theta$ are $(\cos(\theta), \sin(\theta))$. For $\theta = \frac{\pi}{3}$, the coordinates are $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$. So,

$$\cos\left(\frac{\pi}{3}\right) = \frac{1}{2} $$

Determine the tan values of specific angles on the unit circle

We need to determine the $\tan$ values for the angles $30^{\circ}$, $45^{\circ}$, and $60^{\circ}$ on the unit circle:

1. For $30^{\circ}$:

$$\tan 30^{\circ} = \frac{\sin 30^{\circ}}{\cos 30^{\circ}} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$$

2. For $45^{\circ}$:

$$\tan 45^{\circ} = \frac{\sin 45^{\circ}}{\cos 45^{\circ}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$$

3. For $60^{\circ}$:

$$\tan 60^{\circ} = \frac{\sin 60^{\circ}}{\cos 60^{\circ}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$$

What are the sine, cosine, and tangent of the angle $\frac{\pi}{4}$ on the unit circle?

To find the sine, cosine, and tangent of the angle $\frac{\pi}{4}$ on the unit circle, we need to locate the angle on the circle.

The angle $\frac{\pi}{4}$ radians is equivalent to 45 degrees. In the unit circle, this corresponds to the point where both x and y coordinates are equal, as the angle bisects the first quadrant.

The coordinates of the point are $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$.

Therefore:

$$ \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $$

$$ \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} $$

$$ \tan \left( \frac{\pi}{4} \right) = \frac{ \frac{\sqrt{2}}{2} }{ \frac{\sqrt{2}}{2} } = 1 $$

Find the value of csc(π/3) using the unit circle

To find $\csc(\frac{\pi}{3})$, we first need to recall the definition of the cosecant function:

$$\csc(\theta) = \frac{1}{\sin(\theta)}$$

Next, we locate the angle $\frac{\pi}{3}$ on the unit circle. The sine of $\frac{\pi}{3}$ is given by:

$$\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$$

Now, using the definition of cosecant:

$$\csc(\frac{\pi}{3}) = \frac{1}{\sin(\frac{\pi}{3})} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$$

Therefore, $\csc(\frac{\pi}{3}) = \frac{2\sqrt{3}}{3}$.

What are the sine and cosine values of an angle of π/6 on the unit circle?

The angle $\frac{\pi}{6}$ radians corresponds to 30 degrees.

On the unit circle, the coordinates of the point at this angle represent the cosine and sine values.

Therefore, for the angle $\frac{\pi}{6}$:

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

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

Determine the exact values of cotangent for an angle that, when doubled, corresponds to a point on the unit circle where the y-coordinate is equal to the negative square root of 3 divided by 2

To find $\cot(\theta)$, we need to determine the appropriate angle $2\theta$. Given that the y-coordinate of $2\theta$ is $-\frac{\sqrt{3}}{2}$, we know that $2\theta$ corresponds to $240^\circ$ or $300^\circ$ in the unit circle.

1. For $2\theta = 240^\circ$:

$$\theta = \frac{240^\circ}{2} = 120^\circ$$

$$\cot(120^\circ) = \cot(180^\circ – 60^\circ) = -\cot(60^\circ) = -\frac{1}{\sqrt{3}}$$

2. For $2\theta = 300^\circ$:

$$\theta = \frac{300^\circ}{2} = 150^\circ$$

$$\cot(150^\circ) = \cot(180^\circ – 30^\circ) = -\cot(30^\circ) = -\sqrt{3}$$

Therefore, the exact values of $\cot(\theta)$ are $-\frac{1}{\sqrt{3}}$ and $-\sqrt{3}$.

If the unit circle in the complex plane is flipped upside down over the real axis, determine the new coordinates of the point $e^{i\theta}$ where $0 \leq \theta < 2\pi$

Given the point on the unit circle represented by $e^{i\theta}$, we begin by expressing this in terms of its Cartesian coordinates:

\[ e^{i\theta} = \cos(\theta) + i\sin(\theta) \]

When the unit circle is flipped over the real axis, the imaginary part changes sign. Thus, the new coordinates become:

\[ e^{i\theta} \rightarrow \cos(\theta) – i\sin(\theta) \]

Therefore, the new coordinates for the point on the flipped unit circle are:

\[ \boxed{\cos(\theta) – i\sin(\theta)} \]

What is the cosine of the angle 45 degrees on the unit circle?

The angle 45 degrees is equivalent to $\frac{\pi}{4}$ radians.

On the unit circle, the coordinates for an angle of $45^\circ$ or $\frac{\pi}{4}$ radians are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

Thus, the cosine of $45^\circ$ is $\frac{\sqrt{2}}{2}$.

$$ \cos 45^\circ = \frac{\sqrt{2}}{2} $$

Find the cosine of -π/3 using the unit circle

To find the cosine of $-\pi/3$ using the unit circle, follow these steps:

1. Recognize that the angle $-\pi/3$ is a negative angle, which means it is measured clockwise from the positive x-axis.

2. The angle $-\pi/3$ is equivalent to $-60^\circ$.

3. On the unit circle, an angle of $-60^\circ$ corresponds to an angle of $300^\circ$ when measured counterclockwise from the positive x-axis.

4. The coordinates of the point on the unit circle at $300^\circ$ are $(\cos 300^\circ, \sin 300^\circ)$. These coordinates are $(1/2, -\sqrt{3}/2)$.

5. Therefore, the cosine of $-\pi/3$ is the x-coordinate of this point, which is $1/2$.

So, $$\cos(-\pi/3) = \frac{1}{2}$$.