Unit Circle

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:

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

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

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

Finding Specific Tan Values on the Unit Circle

To find the exact $\tan$ values at specific angles on the unit circle, consider the following:

1. $\theta = \frac{\pi}{4}$
At this angle, $\tan(\theta) = \tan\left(\frac{\pi}{4}\right) = 1$

2. $\theta = \frac{2\pi}{3}$
At this angle, $\tan(\theta) = \tan\left(\frac{2\pi}{3}\right) = -\sqrt{3}$

3. $\theta = \frac{7\pi}{6}$
At this angle, $\tan(\theta) = \tan\left(\frac{7\pi}{6}\right) = \frac{1}{\sqrt{3}}$

Find the Real Part of a Complex Number on the Unit Circle

Consider a complex number $z$ on the unit circle in the complex plane. The unit circle can be represented as $|z| = 1$. If $z = e^{i\theta}$, find the real part of $z$.

Solution:

Since $z = e^{i\theta}$, we can use Euler’s formula, which states:

$$e^{i\theta} = \cos \theta + i \sin \theta$$

The real part of $z$ is therefore:

$$ \cos \theta $$

Given a point on a unit circle with coordinates (x, y) and angle θ from the positive x-axis, find the coordinates of the point after rotating by 45 degrees counterclockwise

Given the initial coordinates $(x, y)$ and angle $\theta$, the coordinates after rotating by $45^\circ$ counterclockwise can be found using the rotation matrix:

$$ \begin{bmatrix} \cos(45^\circ) & -\sin(45^\circ) \\ \sin(45^\circ) & \cos(45^\circ) \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$

Since $\cos(45^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2}$, the formula becomes:

$$ \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $$

Performing the matrix multiplication, we get:

$$ \begin{bmatrix} \frac{\sqrt{2}}{2}x – \frac{\sqrt{2}}{2}y \\ \frac{\sqrt{2}}{2}x + \frac{\sqrt{2}}{2}y \end{bmatrix} $$

Thus, the new coordinates are:

$$ \left( \frac{\sqrt{2}}{2}(x – y), \frac{\sqrt{2}}{2}(x + y) \right) $$

In which quadrant does the angle lie on the unit circle?

$$Given \ an \ angle \ of \ 150^{\circ}, \ we \ need \ to \ determine \ which \ quadrant \ it \ lies \ in.$$

$$Quadrant \, I: \ 0^{\circ} \leq \theta < 90^{\circ}$$

$$Quadrant \, II: \ 90^{\circ} \leq \theta < 180^{\circ}$$

$$Quadrant \, III: \ 180^{\circ} \leq \theta < 270^{\circ}$$

$$Quadrant \, IV: \ 270^{\circ} \leq \theta < 360^{\circ}$$

$$Since \ 150^{\circ} \ lies \ between \ 90^{\circ} \ and \ 180^{\circ}, \ it \ is \ in \ Quadrant \, II.$$

Determine the coordinates on the unit circle for 150 degrees and the corresponding angles in radians

First, convert $150^\circ$ to radians:

$$\theta = 150^\circ \times \frac{\pi}{180^\circ} = \frac{5\pi}{6}$$

Next, use the radian measure to find the coordinates on the unit circle. The coordinates for an angle of $\frac{5\pi}{6}$ are:

$$\left(\cos\left(\frac{5\pi}{6}\right), \sin\left(\frac{5\pi}{6}\right)\right) = \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$$

Thus, the coordinates for $150^\circ$ are $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.

Find the coordinates on the unit circle for an angle of \(\frac{\pi}{4}\)

To find the coordinates on the unit circle for an angle of $\frac{\pi}{4}$, we need to use the unit circle definition.

The unit circle has a radius of 1, and the coordinates for any angle $\theta$ can be found using $\cos(\theta)$ and $\sin(\theta)$.

For $\theta = \frac{\pi}{4}$, we have:

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

and

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

Thus, the coordinates are:

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