Homework

What are the coordinates of the point on the unit circle corresponding to an angle of 45 degrees?

To determine the coordinates of the point on the unit circle at $45^\circ$, we use the fact that the unit circle has a radius of 1 and the coordinates are given by $ (\cos \theta, \sin \theta) $.

For $\theta = 45^\circ$, we have:

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

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

Therefore, the coordinates are:

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

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

Let’s find the coordinates of a point on the unit circle corresponding to an angle of $\frac{5\pi}{4}$ radians.

The unit circle equation is given by:

$$x^2 + y^2 = 1$$

For an angle $\theta$, the coordinates $(x, y)$ are given by:

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

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

Substituting $\theta = \frac{5\pi}{4}$:

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

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

Hence, the coordinates of the point are:

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

Find the value and angle for the given csc value

Given $csc(\theta) = \frac{5}{3}$, find the corresponding angle $\theta$.

We know:

$$csc(\theta) = \frac{1}{sin(\theta)}$$

Given,

$$\frac{1}{sin(\theta)} = \frac{5}{3}$$

So,

$$sin(\theta) = \frac{3}{5}$$

To find $\theta$, we take the inverse sine:

$$\theta = sin^{-1}(\frac{3}{5})$$

Using a calculator, we find:

$$\theta \approx 36.87^\circ \, or \, \theta \approx 143.13^\circ$$

Find the value of the cosecant function for an angle in the unit circle

Answer 1:

Given an angle \( \theta \) in the unit circle, we need to find the value of \( \csc(\theta) \). Recall that \( \csc(\theta) = \frac{1}{\sin(\theta)} \).

Let’s consider \( \theta = \frac{5\pi}{6} \). First, we find \( \sin\left(\frac{5\pi}{6}\right) \). Since \( \sin\left(\frac{5\pi}{6}\right) = \sin\left(\pi – \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) \), we have \( \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} \).

Thus, \( \csc\left(\frac{5\pi}{6}\right) = \frac{1}{\sin\left(\frac{5\pi}{6}\right)} = \frac{1}{\frac{1}{2}} = 2 \).

Find the value of cotangent at $\frac{\pi}{4}$ on the unit circle

To find $\cot \left( \frac{\pi}{4} \right)$, we use the definition of cotangent in terms of sine and cosine.

$$\cot \theta = \frac{\cos \theta}{\sin \theta}$$

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

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

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

Therefore,

$$\cot \left( \frac{\pi}{4} \right) = \frac{\cos \left( \frac{\pi}{4} \right)}{\sin \left( \frac{\pi}{4} \right)} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$$

Calculate the exact value of cos(5π/6) using the unit circle

We must first determine the reference angle for $ \frac{5\pi}{6} $. This angle is in the second quadrant.

The reference angle for $ \frac{5\pi}{6} $ is $ \pi – \frac{5\pi}{6} = \frac{\pi}{6} $.

In the second quadrant, the cosine function is negative. Thus,

$$ \cos(\frac{5\pi}{6}) = -\cos(\frac{\pi}{6}) $$

We know that $ \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} $, therefore,

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

Find the Equations of Circles on the Unit Circle

To find the equations of all circles on the unit circle, we start with the general form of a circle’s equation:

$$ (x – h)^2 + (y – k)^2 = r^2$$

Since we are dealing with the unit circle, the radius r is 1. Thus, the equation simplifies to:

$$ (x – h)^2 + (y – k)^2 = 1$$

Here, (h, k) represents the center of the circle. Because the unit circle is centered at the origin (0, 0), h and k are both 0. Therefore, the equation of the unit circle is:

$$ x^2 + y^2 = 1$$

Find the value of tan(θ) where θ is a special angle on the unit circle

To find the value of $\tan(\theta)$ where $\theta$ is a special angle on the unit circle, we use the definition $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.

For $\theta = \frac{\pi}{4}$, the sine and cosine values are both $\frac{\sqrt{2}}{2}$.

Therefore, $\tan(\frac{\pi}{4}) = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$.

Find the values of sin(π/4) and cos(π/4) using the unit circle

Using the unit circle, we can find the values of $\sin(\frac{\pi}{4})$ and $\cos(\frac{\pi}{4})$ by locating the angle $\frac{\pi}{4}$ radians. This angle corresponds to a 45-degree angle in the unit circle.

At this angle, both the x-coordinate (which represents $\cos(\frac{\pi}{4})$) and the y-coordinate (which represents $\sin(\frac{\pi}{4})$) are equal to $\frac{\sqrt{2}}{2}$.

Thus,

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

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