Homework

Find the cosine of an angle using the unit circle in the complex plane

Given an angle \( \theta \) in the complex plane, the unit circle can be used to find the cosine of the angle. The cosine of the angle \( \theta \) is the x-coordinate of the point where the terminal side of the angle intersects the unit circle.

Let’s consider \( \theta = \frac{\pi}{4} \), find \( \cos(\theta) \).

On the unit circle, the coordinates of the point at angle \( \frac{\pi}{4} \) are \( \left( \cos \left( \frac{\pi}{4} \right), \sin \left( \frac{\pi}{4} \right) \right) \).

Since \( \frac{\pi}{4} \) is a 45-degree angle, the coordinates are \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \).

Thus, the cosine of \( \frac{\pi}{4} \) is:

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

Convert the point on the unit circle given in Cartesian coordinates (sqrt(3)/2, 1/2) to its corresponding angle in degrees and radians, and verify the solution by converting the angle back to Cartesian coordinates

We are given the point $(\sqrt{3}/2, 1/2)$ on the unit circle. To find the corresponding angle, we use the following trigonometric relationships:

$$x = \cos(\theta)$$

$$y = \sin(\theta)$$

Thus, we have:

$$\cos(\theta) = \sqrt{3}/2$$

$$\sin(\theta) = 1/2$$

For the angle $\theta$ that satisfies these equations, we recognize that these are standard values. The angle $\theta$ is $30^{\circ}$ or $\pi/6$ radians.

To verify, we will convert $30^{\circ}$ back to Cartesian coordinates:

$$\cos(30^{\circ}) = \sqrt{3}/2, \sin(30^{\circ}) = 1/2$$

Thus, the point $(\cos(30^{\circ}), \sin(30^{\circ})) = (\sqrt{3}/2, 1/2)$ matches the given point. Therefore, $\theta = 30^{\circ}$ or $\pi/6$ radians.

Given a point on the unit circle, determine the coordinates and verify the trigonometric identities

Let’s consider a point $P(\cos\theta, \sin\theta)$ on the unit circle where $\theta = \frac{5\pi}{6}$. To find the coordinates and verify trigonometric identities:

First, we calculate the coordinates:

$$P = (\cos \frac{5\pi}{6}, \sin \frac{5\pi}{6})$$

Using the unit circle, we know:

$$\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}$$

$$\sin \frac{5\pi}{6} = \frac{1}{2}$$

Thus, the coordinates are:

$$P = \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$$

Next, we verify the Pythagorean identity:

$$\cos^2 \theta + \sin^2 \theta = 1$$

Substituting in the values, we get:

$$\left(-\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 = \frac{3}{4} + \frac{1}{4} = 1$$

Which confirms that the point lies on the unit circle.

What is the cosine of the angle π/3 on the unit circle?

To find the cosine of the angle \( \frac{\pi}{3} \) on the unit circle, we need to locate this angle on the circle.

The angle \( \frac{\pi}{3} \) corresponds to 60 degrees.

On the unit circle, the coordinates of the point at angle \( \frac{\pi}{3} \) are \( \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) \).

The cosine of an angle is the x-coordinate of the corresponding point on the unit circle.

Therefore, \( \cos \left( \frac{\pi}{3} \right) = \frac{1}{2} \).

Find all angles θ between 0 and 2π such that cos(θ) = -1/2

To find the angles $\theta$ such that $\cos(\theta) = -\frac{1}{2}$, we start by identifying the quadrants where $\cos(\theta)$ is negative. Cosine is negative in the second and third quadrants.

First, we find the reference angle:

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

Now, we find the angles in the second and third quadrants:

Second quadrant: $$\pi – \frac{\pi}{3} = \frac{2\pi}{3}$$

Third quadrant: $$\pi + \frac{\pi}{3} = \frac{4\pi}{3}$$

Thus, the angles are $\frac{2\pi}{3}$ and $\frac{4\pi}{3}$.

Find the coordinates of a point on the unit circle at an angle of 45 degrees from the positive x-axis

To find the coordinates of a point on the unit circle, we use the trigonometric functions sine and cosine.

The angle given is $45^{\circ}$.

Using the unit circle properties:

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

$$y = \sin 45^{\circ} = \frac{\sqrt{2}}{2}$$

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

Find the value of x such that cos(x) = -1/2 and sin(x) is negative on the unit circle

To solve for $x$ such that $\cos(x) = -\frac{1}{2}$ and $\sin(x)$ is negative on the unit circle, follow these steps:

1. Identify the angles where $\cos(x) = -\frac{1}{2}$. This occurs at $x = \frac{2\pi}{3}$ and $x = \frac{4\pi}{3}$ in radians.

2. Determine which of these angles has a negative sine value. The sine function is negative in the third and fourth quadrants.

3. Since $\frac{2\pi}{3}$ is in the second quadrant and $\frac{4\pi}{3}$ is in the third quadrant, we choose $x = \frac{4\pi}{3}$.

Therefore, the value of $x$ is:

$$ x = \frac{4\pi}{3} $$

Find the exact values of sine and cosine for the angle 5π/4 using the unit circle

To find the exact values of sine and cosine for the angle $\frac{5\pi}{4}$, we start by determining in which quadrant the angle lies.

The angle $\frac{5\pi}{4}$ is in the third quadrant because $\frac{5\pi}{4} > \pi$ and $\frac{5\pi}{4} < \frac{3\pi}{2}$.

In the third quadrant, both sine and cosine are negative.

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

We know that $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ and $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$.

Hence, for the third quadrant:

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

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

Given the point P on the unit circle at an angle of 210 degrees, find cos(210°) and sin(210°)

To find $\cos(210^{\circ})$ and $\sin(210^{\circ})$, we start by converting the angle to radians:

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

The reference angle for $\frac{7\pi}{6}$ is $30^{\circ}$ or $\frac{\pi}{6}$.

The coordinates of a point on the unit circle at an angle $\theta$ in the third quadrant are $(-\cos(\theta), -\sin(\theta))$.

Since $\cos(30^{\circ}) = \frac{\sqrt{3}}{2}$ and $\sin(30^{\circ}) = \frac{1}{2}$, we get:

$$\cos(210^{\circ}) = -\cos(30^{\circ}) = -\frac{\sqrt{3}}{2}$$

$$\sin(210^{\circ}) = -\sin(30^{\circ}) = -\frac{1}{2}$$

Therefore, the coordinates of the point are:

$$P(\cos(210^{\circ}), \sin(210^{\circ})) = \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$$

Find the coordinates of the points where the unit circle intersects the x-axis

$$\text{The unit circle has the equation } x^2 + y^2 = 1.$$

$$\text{To find the intersection with the x-axis, we set } y = 0.$$

$$x^2 + 0^2 = 1$$

$$x^2 = 1$$

$$x = \pm 1.$$

$$\text{Thus, the coordinates are } (1, 0) \text{ and } (-1, 0).$$