Unit Circle

Finding the Tangent of Angles on the Unit Circle

To find the tangent of an angle θ on the unit circle, we use the definition of tangent in terms of sine and cosine: $$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$

Consider the angle θ = 45 degrees. The coordinates on the unit circle are (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}). Therefore,

$$\tan(45°) = \frac{\sin(45°)}{\cos(45°)} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 $$

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 $\frac{\pi}{6}$:

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

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

Convert an angle from degrees to radians using the unit circle

The formula to convert degrees to radians is: \( \theta = \frac{\pi}{180} \times \text{degrees} \)

Given an angle of 120 degrees, we use the following calculation:

$$ \theta = \frac{\pi}{180} \times 120 $$

This simplifies to:

$$ \theta = \frac{2\pi}{3} $$

Hence, 120 degrees is equivalent to \( \frac{2\pi}{3} \) radians.

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 $-\sqrt{3}$.

Find the value of tan(4π/3) on the unit circle

To find $ \tan \left( \frac{4\pi}{3} \right) $ on the unit circle, we note that $ \frac{4\pi}{3} $ radians is in the third quadrant.

In the third quadrant, both sine and cosine are negative. The reference angle for $ \frac{4\pi}{3} $ is $ \frac{\pi}{3} $.

We know that:

$$ \tan \left( \frac{\pi}{3} \right) = \sqrt{3} $$

Since tangent is positive in the third quadrant:

$$ \tan \left( \frac{4\pi}{3} \right) = \tan \left( \pi + \frac{\pi}{3} \right) = \tan \left( \frac{\pi}{3} \right) = \sqrt{3} $$

Therefore, $ \tan \left( \frac{4\pi}{3} \right) = \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 $\frac{\sqrt{2}}{2}$.

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

Calculate tan(4π/3) using the Unit Circle

First, we need to find the reference angle for $\frac{4\pi}{3}$. The angle $\frac{4\pi}{3}$ is in the third quadrant.

The reference angle is $$\pi – (\frac{4\pi}{3} – \pi) = \frac{\pi}{3}$$.

In the third quadrant, the tangent function is positive, so we have:

$$\tan(\frac{4\pi}{3}) = \tan(\frac{\pi}{3})$$

$$\tan(\frac{\pi}{3}) = \sqrt{3}$$

So, the answer is $$\sqrt{3}$$.

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

Find the angle whose cosine is -2/3 on the unit circle

Given that $\cos(\theta) = -\frac{2}{3}$, we need to find the angle $\theta$ on the unit circle.

Since cosine represents the x-coordinate on the unit circle, we look for the angle in the second and third quadrants where the cosine values are negative.

In the second quadrant, we have:

$$ \theta = \pi – \arccos\left(\frac{2}{3}\right) $$

In the third quadrant, we have:

$$ \theta = \pi + \arccos\left(\frac{2}{3}\right) $$

Therefore, the angles whose cosine is $-\frac{2}{3}$ are:

$$ \theta = \pi – \arccos\left(\frac{2}{3}\right) \text{ and } \theta = \pi + \arccos\left(\frac{2}{3}\right) $$