Math

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:

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

Find the cosine value for a given angle on the unit circle

Consider an angle $\theta = \frac{\pi}{3}$ on the unit circle.

We know from trigonometry that the point corresponding to $\theta = \frac{\pi}{3}$ has coordinates $(\cos(\frac{\pi}{3}), \sin(\frac{\pi}{3}))$.

Using the unit circle values, we find

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

Therefore, the cosine of $\frac{\pi}{3}$ is $\frac{1}{2}$.

Find the coordinates and angle measure for the point on the unit circle where the secant function is undefined

Let’s start by identifying where the secant function is undefined. Recall that $\sec(\theta) = \frac{1}{\cos(\theta)}$.

The secant function is undefined when $\cos(\theta) = 0$. On the unit circle, this occurs at the points where the x-coordinate is 0. These points are at $\theta = \frac{\pi}{2}$ and $\theta = \frac{3\pi}{2}$.

At $\theta = \frac{\pi}{2}$, the coordinates are $(0, 1)$.

At $\theta = \frac{3\pi}{2}$, the coordinates are $(0, -1)$.

Thus, the secant function is undefined at $\theta = \frac{\pi}{2}$ and $\theta = \frac{3\pi}{2}$, with corresponding coordinates $(0, 1)$ and $(0, -1)$ respectively.

Strategies to Easily Learn the Unit Circle

To understand the unit circle, consider the following:

1. Identify the key points on the unit circle where the angle is 0, $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$, and $\frac{\pi}{2}$.

2. Recall the coordinates of these points: $ (1, 0)$, $ \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$, $ \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, $ \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$, and $(0, 1)$.

3. Notice the symmetry in the unit circle. For instance, $ \sin(\theta) = \cos(\frac{\pi}{2} – \theta)$.

4. Practice by drawing the unit circle and labeling these points.

5. Use the Pythagorean identity $ \sin^2(\theta) + \cos^2(\theta) = 1$ to verify positions.

Answer: Understanding the coordinates and symmetry of key angles helps in mastering the unit circle.

Find the values of sine, cosine, and tangent for an angle of 120 degrees using the unit circle

To find the values of $\sin$, $\cos$, and $\tan$ for an angle of 120 degrees, first convert the angle to radians:

$$120^\circ = \frac{120 \pi}{180} = \frac{2 \pi}{3}$$

Next, locate the angle on the unit circle. The angle $\frac{2 \pi}{3}$ is in the second quadrant, where the sine is positive, and the cosine and tangent are negative.

The reference angle for $120^\circ$ is $180^\circ – 120^\circ = 60^\circ$.

For $60^\circ$, we have:

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

$$\cos 60^\circ = \frac{1}{2}$$

Since 120 degrees is in the second quadrant:

$$\sin 120^\circ = \sin 60^\circ = \frac{\sqrt{3}}{2}$$

$$\cos 120^\circ = -\cos 60^\circ = -\frac{1}{2}$$

$$\tan 120^\circ = \frac{\sin 120^\circ}{\cos 120^\circ} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3}$$