Math

What are the sine and cosine values of 45 degrees on the unit circle?

First, let’s convert 45 degrees into radians using the conversion factor $\pi / 180$.

$$\text{Radians} = 45 \times \frac{\pi}{180} = \frac{\pi}{4}$$

On the unit circle, the coordinates corresponding to an angle of $\frac{\pi}{4}$ radians are given by $\left(\cos \frac{\pi}{4}, \sin \frac{\pi}{4}\right)$.

We know that:

$$\cos \frac{\pi}{4} = \cos 45^{\circ} = \frac{\sqrt{2}}{2}$$

$$\sin \frac{\pi}{4} = \sin 45^{\circ} = \frac{\sqrt{2}}{2}$$

Therefore, the sine and cosine values for 45 degrees on the unit circle are $\frac{\sqrt{2}}{2}$ and $\frac{\sqrt{2}}{2}$ respectively.

Find the value of sin(θ) for θ = 7π/6 using the unit circle

To find $\sin(\theta)$ for $\theta = \frac{7\pi}{6}$, we need to locate the angle on the unit circle.

First, note that $\frac{7\pi}{6}$ is in the third quadrant where sine is negative.

$\frac{7\pi}{6}$ is $30^\circ$ past $\pi$ (180 degrees).

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

In the third quadrant, the sine of $\frac{\pi}{6}$ is $-\frac{1}{2}$.

Thus, $\sin(\frac{7\pi}{6}) = -\frac{1}{2}$.

Identify the quadrant of an angle

Given an angle of $135°$, determine the quadrant in which this angle lies on the unit circle.

The unit circle is divided into four quadrants:

Quadrant I: $0°$ to $90°$

Quadrant II: $90°$ to $180°$

Quadrant III: $180°$ to $270°$

Quadrant IV: $270°$ to $360°$

Since $135°$ is greater than $90°$ and less than $180°$,

$$135°$$

lies in Quadrant II.

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

To find the value of $\tan(\theta)$ on the unit circle, we need to use the relationship $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.

Let’s consider $\theta = \frac{\pi}{4}$. On the unit circle, $\sin\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$.

Thus,

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

So, $\tan\left(\frac{\pi}{4}\right) = 1$.

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:

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