Homework

Determine the sine and cosine of the angle π/4 on the unit circle

To find the sine and cosine of the angle $\frac{\pi}{4}$ on the unit circle, we use the definitions of sine and cosine for the unit circle.

For an angle $\theta$ in the unit circle, $\cos(\theta)$ is the x-coordinate and $\sin(\theta)$ is the y-coordinate of the corresponding point.

At $\theta = \frac{\pi}{4}$, the coordinates are known to be $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.

Thus,

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

and

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

Find the value of \( \theta \) if \( \tan(\theta) = 2 \) and \( \theta \) is in the second quadrant Then, calculate the coordinates of the corresponding point on the unit circle

We know that \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). In the second quadrant, sine is positive and cosine is negative. Let us find \( \theta \) such that \( \tan(\theta) = 2 \). The reference angle \( \theta_r \) is given by:

$$ \theta_r = \arctan(2) $$

Since \( \theta \) is in the second quadrant, the angle \( \theta \) is:

$$ \theta = \pi – \theta_r = \pi – \arctan(2) $$

Next, to find the coordinates of the corresponding point on the unit circle, we use the unit circle property \((\cos(\theta), \sin(\theta))\). First we find \( \sin(\theta) \) and \( \cos(\theta) \) using:

$$ \sin(\theta) = \frac{2}{\sqrt{1 + 2^2}} = \frac{2}{\sqrt{5}} \quad \text{and} \quad \cos(\theta) = -\frac{1}{\sqrt{1 + 2^2}} = -\frac{1}{\sqrt{5}} $$

Therefore, the coordinates are:

$$ \left( -\frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}} \right) $$

Given a point P on the unit circle, find the coordinates of P if the angle formed with the positive x-axis is θ, where θ satisfies 0 ≤ θ ≤ 2π and the coordinates satisfy the equation of the circle x^2 + y^2 = 1 Provide three different coordinate sets for

$$\theta = \frac{\pi}{6}$$

For $\theta = \frac{\pi}{6}$, the coordinates $(x,y)$ on the unit circle are given by:

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

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

Thus, the coordinates are $$\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$$.

$$\theta = \frac{\pi}{4}$$

For $\theta = \frac{\pi}{4}$, the coordinates $(x,y)$ on the unit circle are given by:

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

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

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

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

For $\theta = \frac{\pi}{3}$, the coordinates $(x,y)$ on the unit circle are given by:

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

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

Thus, the coordinates are $$\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$$.

Find the general solution for \( \theta \) in the equation \( \sin(\theta) = \frac{1}{2} \) using the unit circle

To find the general solution for the equation $\sin(\theta) = \frac{1}{2}$, we use the unit circle. On the unit circle, $\sin(\theta) = \frac{1}{2}$ at $\theta = \frac{\pi}{6}$ and $\theta = \frac{5\pi}{6}$ within one period $[0, 2\pi)$.

Therefore, the general solutions are:

$$\theta = \frac{\pi}{6} + 2k\pi, \quad k \in \mathbb{Z}$$

and

$$\theta = \frac{5\pi}{6} + 2k\pi, \quad k \in \mathbb{Z}$$

Find the coordinates of a point on the unit circle at an angle of 45 degrees

To find the coordinates of a point on the unit circle at an angle of $45^{\circ}$, we use the sine and cosine functions.

First, we convert the angle to radians:

$$45^{\circ} = \frac{45 \pi}{180} = \frac{\pi}{4}$$

Now, we find the sine and cosine of $\frac{\pi}{4}$:

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

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

Therefore, the coordinates of the point are:

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

Find the sine and cosine values for an angle of 5π/4 radians on the unit circle

To find the sine and cosine values for an angle of $\frac{5\pi}{4}$ radians, we need to locate this angle on the unit circle.

First, we recognize that $\frac{5\pi}{4}$ radians is in the third quadrant because it is more than $\pi$ radians (180 degrees) but less than $\frac{3\pi}{2}$ radians (270 degrees).

The reference angle for $\frac{5\pi}{4}$ radians is $\frac{5\pi}{4} – \pi = \frac{\pi}{4}$ radians.

In the third quadrant, the sine and cosine values are both negative. For the reference angle $\frac{\pi}{4}$, the sine and cosine values are both $\frac{\sqrt{2}}{2}$.

Therefore, the sine and cosine values for $\frac{5\pi}{4}$ radians are:

$$\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$

$$\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$

Determine the Quadrant of Multiple Angles on the Unit Circle

Given the angles $30^\circ$, $150^\circ$, and $240^\circ$, determine the quadrant each angle lies in on the unit circle.

1. For $30^\circ$, it is in the first quadrant because it is between $0^\circ$ and $90^\circ$.

2. For $150^\circ$, it is in the second quadrant because it is between $90^\circ$ and $180^\circ$.

3. For $240^\circ$, it is in the third quadrant because it is between $180^\circ$ and $270^\circ$.

Therefore, $30^\circ$ lies in the first quadrant, $150^\circ$ lies in the second quadrant, and $240^\circ$ lies in the third quadrant.

Determine the coordinates of a point on the unit circle given a specific angle

Given an angle \( \theta = \frac{5\pi}{6} \), find the coordinates of the corresponding point on the unit circle.

1. The angle \( \frac{5\pi}{6} \) is in the second quadrant where sine is positive and cosine is negative.

2. Using the unit circle, the coordinates for \( \theta = \frac{5\pi}{6} \) can be found using the reference angle \( \pi – \frac{5\pi}{6} = \frac{\pi}{6} \).

3. The coordinates for \( \frac{\pi}{6} \) are \( \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) \).

4. Since \( \frac{5\pi}{6} \) is in the second quadrant, the x-coordinate will be negative.

Hence, the coordinates are $$\left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right)$$.

Determine the quadrant of point P on the unit circle where P has coordinates (cos(θ), sin(θ)), given that θ = 210 degrees

First, convert the given angle to radians:

$$\theta = 210^\circ \times \frac{\pi}{180^\circ} = \frac{7\pi}{6}$$

Determine the coordinates of point P:

$$P = (\cos(210^\circ), \sin(210^\circ))$$

Since $210^\circ$ is in the third quadrant, $\cos(210^\circ)$ is negative and $\sin(210^\circ)$ is also negative. Therefore, the point P is in the third quadrant.

The answer is: The point P is in the third quadrant.