Unit Circle

Find the coordinates on the Unit Circle

To determine the coordinates on the unit circle corresponding to an angle of $ \frac{5\pi}{6} $, we use the trigonometric functions sine and cosine.

The cosine of $ \frac{5\pi}{6} $ corresponds to the x-coordinate, and the sine of $ \frac{5\pi}{6} $ corresponds to the y-coordinate.

Calculating these values:

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

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

So, the coordinates are:

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

Find the exact value of the trigonometric functions for the angle θ = 5π/6 using the unit circle

First, locate the angle $\theta = \frac{5\pi}{6}$ on the unit circle. This angle is in the second quadrant.

In the second quadrant, sine is positive and cosine is negative.

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

From the unit circle, $\sin(\frac{\pi}{6}) = \frac{1}{2}$ and $\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$.

Therefore, $\sin(\frac{5\pi}{6}) = \sin(\frac{\pi}{6}) = \frac{1}{2}$ and $\cos(\frac{5\pi}{6}) = -\cos(\frac{\pi}{6}) = -\frac{\sqrt{3}}{2}$.

Hence, the exact values are:

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

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

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

Find the values of tan(θ) for specific angles on the unit circle

$$For \ θ = \frac{3π}{4}, \ we \ know \ that \ tan(θ) = \frac{sin(θ)}{cos(θ)}$$

$$sin(θ) = sin(\frac{3π}{4}) = \frac{1}{\sqrt{2}}, \ cos(θ) = cos(\frac{3π}{4}) = -\frac{1}{\sqrt{2}}$$

$$Therefore, \ tan(θ) = \frac{\frac{1}{\sqrt{2}}}{-\frac{1}{\sqrt{2}}} = -1$$

Find the value of tan for the angle 45 degrees on the unit circle

To find the value of $\tan 45^{\circ}$ on the unit circle, we use the definition of $\tan$:

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

For $\theta = 45^{\circ}$, we know that $\sin 45^{\circ} = \frac{\sqrt{2}}{2}$ and $\cos 45^{\circ} = \frac{\sqrt{2}}{2}$.

Substituting these values in, we get:

$$\tan 45^{\circ} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$$

So, the value of $\tan 45^{\circ}$ is 1.

Find the coordinates of the point on the unit circle where the angle is 5π/4 radians

To find the coordinates of the point on the unit circle where the angle is $ \frac{5\pi}{4} $ radians, we can use the definitions of sine and cosine for the unit circle.

The angle $ \frac{5\pi}{4} $ is in the third quadrant, where both sine and cosine are negative.

For the unit circle, the coordinates are given by $(\cos \theta, \sin \theta)$.

Thus, we find:

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

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

Therefore, the coordinates are:

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

Find the coordinates of the point on the unit circle at an angle of 5π/4 radians

To find the coordinates of the point on the unit circle at an angle of $\frac{5\pi}{4}$ radians, we can use the cosine and sine functions:

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

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

First, let’s calculate the cosine value:

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

Next, let’s calculate the sine value:

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

Therefore, the coordinates of the point are:

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

Find the sine and cosine values for the angle 5π/6 on the unit circle, and determine the corresponding point on the unit circle

First, we need to determine the reference angle for $\frac{5\pi}{6}$. The reference angle is $\pi – \frac{5\pi}{6} = \frac{\pi}{6}$.

In the second quadrant, sine is positive and cosine is negative.

The sine value for $\frac{\pi}{6}$ is $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$.

The cosine value for $\frac{\pi}{6}$ is $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$.

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

The corresponding point on the unit circle is $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.

Find the values of sine and cosine for an angle on the unit circle

Given an angle $\theta$, find the values of $\sin(\theta)$ and $\cos(\theta)$ using the unit circle equation. Suppose $\theta = \frac{\pi}{4}$.

The unit circle equation is given by:

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

For $\theta = \frac{\pi}{4}$, the corresponding point on the unit circle is $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.

Therefore,

$$\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \quad \text{and} \quad \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}.$$

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

The angle $\frac{\pi}{3}$ corresponds to 60 degrees on the unit circle.

The coordinates of this angle on the unit circle are $(\frac{1}{2}, \frac{\sqrt{3}}{2})$.

Therefore, the cosine of the angle $\frac{\pi}{3}$ is $\frac{1}{2}$, and the sine is $\frac{\sqrt{3}}{2}$.

Find the coordinates on the unit circle for an angle of 5π/6

To find the coordinates of the point on the unit circle corresponding to an angle of $\frac{5\pi}{6}$, we can use the sine and cosine functions.

The angle $\frac{5\pi}{6}$ is in the second quadrant.

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

We know that for $\frac{\pi}{6}$:

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

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

Since $\frac{5\pi}{6}$ is in the second quadrant, the x-coordinate (cosine) will be negative and the y-coordinate (sine) will be positive.

Therefore, the coordinates are:

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