Unit Circle

Finding the Coordinates of a Point on the Unit Circle

Given an angle of $\theta = \frac{\pi}{3}$ radians, find the coordinates of the corresponding point on the unit circle.

First, recall the unit circle definition: for any angle $\theta$, the coordinates of the point on the unit circle are given by $(\cos \theta, \sin \theta)$. For $\theta = \frac{\pi}{3}$:

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

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

Thus, the coordinates are:

$$(\frac{1}{2}, \frac{\sqrt{3}}{2})$$

Find the sine and cosine values for the angle \(\theta = 45^{\circ}\) on the unit circle

To find the sine and cosine values for the angle $\theta = 45^{\circ}$ on the unit circle:

1. Note that $\theta = 45^{\circ}$ is in the first quadrant.

2. The coordinates of the corresponding point on the unit circle are given by $(\cos(45^{\circ}), \sin(45^{\circ}))$.

3. Using standard values:

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

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

Thus, the sine and cosine values for $\theta = 45^{\circ}$ are both $\frac{\sqrt{2}}{2}$.

Given that \( \csc(\theta) = 2 \) and \( \theta \) lies in the second quadrant, find the exact value of \( \theta \) and verify using trigonometric identities

Given:

$$ \csc(\theta) = 2 $$

Since \( \csc(\theta) = \frac{1}{\sin(\theta)} \), we get:

$$ \sin(\theta) = \frac{1}{2} $$

In the second quadrant, angle \( \theta \) where \( \sin(\theta) = \frac{1}{2} \) is:

$$ \theta = 180^\circ – 30^\circ = 150^\circ $$

Converting to radians:

$$ \theta = \pi – \frac{\pi}{6} = \frac{5\pi}{6} $$

Verification:

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

Thus, the exact value of \( \theta \) is:

$$ \boxed{\frac{5\pi}{6}} $$

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:

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