Unit Circle

Calculate the exact values of sin and cos at θ = 5π/6

To find the exact values of $ \sin $ and $ \cos $ at $ \theta = \frac{5\pi}{6} $, we use the unit circle.

First, find the reference angle:

$$ \theta_{ref} = \pi – \frac{5\pi}{6} = \frac{\pi}{6} $$

Using the reference angle $ \frac{\pi}{6} $, we know the exact values for sine and cosine are:

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

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

Since $ \theta = \frac{5\pi}{6} $ is in the second quadrant, sine is positive, and cosine is negative.

Therefore:

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

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

Find the tangent of an angle on a unit circle

Given an angle $ \theta $ on a unit circle, the tangent of the angle is defined as the ratio of the sine to the cosine of the angle.

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

For instance, if $ \theta = \frac{\pi}{4} $:

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

Thus, the tangent of $ \frac{\pi}{4} $ is 1.

Find the sine and cosine values at specific angles on the unit circle

To find the sine and cosine values at specific angles on the unit circle, we use the definitions of sine and cosine in terms of the unit circle.

For example, at an angle of 30 degrees (or $\frac{\pi}{6}$ radians):

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

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

Find the coordinates of a point on the unit circle that corresponds to an angle of 7π/4

To find the coordinates of the point on the unit circle that corresponds to an angle of $\frac{7\pi}{4}$, we use the sine and cosine functions:

$$x = \cos\left(\frac{7\pi}{4}\right)$$

$$y = \sin\left(\frac{7\pi}{4}\right)$$

Since $\frac{7\pi}{4}$ is in the fourth quadrant, we have:

$$\cos\left(\frac{7\pi}{4}\right) = \frac{1}{\sqrt{2}}$$

$$\sin\left(\frac{7\pi}{4}\right) = -\frac{1}{\sqrt{2}}$$

So, the coordinates are:

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

Determine the tangent values for the primary angles on the unit circle

To determine the tangent values for the primary angles on the unit circle, we need to evaluate the tangent function at $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$ and $2\pi$.

$$ \text{tan}(0) = 0 $$

$$ \text{tan}\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} $$

$$ \text{tan}\left(\frac{\pi}{4}\right) = 1 $$

$$ \text{tan}\left(\frac{\pi}{3}\right) = \sqrt{3} $$

$$ \text{tan}\left(\frac{\pi}{2}\right) = \text{undefined} $$

$$ \text{tan}(\pi) = 0 $$

$$ \text{tan}\left(\frac{3\pi}{2}\right) = \text{undefined} $$

$$ \text{tan}(2\pi) = 0 $$

How to find sine, cosine, and tangent for an angle using the unit circle?

To find the sine, cosine, and tangent of an angle using the unit circle, follow these steps:

1. Locate the angle on the unit circle.

2. Identify the coordinates $(x, y)$ of the point where the terminal side of the angle intersects the unit circle.

3. The coordinates correspond to $\cos(\theta)$ and $\sin(\theta)$ respectively:

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

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

4. Calculate $\tan(\theta)$ as follows:

$$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{y}{x} $$

Determine the coordinates of the point where the angle θ = π/3 on the unit circle

First, recall that the unit circle has a radius of 1. For the angle $ \theta = \frac{\pi}{3} $, we use the definitions of sine and cosine:

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

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

When $ \theta = \frac{\pi}{3} $, we have:

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

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

Therefore, the coordinates of the point are:

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

Find the values of sin(θ) and cos(θ) where θ is 5π/4 radians on the unit circle

Given $\theta = \frac{5\pi}{4}$, we need to find the values of $\sin(\theta)$ and $\cos(\theta)$ on the unit circle.

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

In the third quadrant, for an angle of $\frac{5\pi}{4}$,

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

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

Find the coordinates of the point on the unit circle where the angle is 5π/6

To find the coordinates of the point on the unit circle where the angle is $\frac{5\pi}{6}$, we use the unit circle trigonometric identities for sine and cosine.

Since $\frac{5\pi}{6}$ is in the second quadrant:

The x-coordinate is:

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

The y-coordinate is:

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

So, the coordinates are:

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

Find the coordinates of the point on the unit circle for an angle of 3π/4 radians

To find the coordinates of the point on the unit circle for an angle of $ \frac{3\pi}{4} $ radians, we need to use the unit circle definition:

For an angle $ \theta $, the coordinates are given by:

$$ (\cos(\theta), \sin(\theta)) $$

Here, $ \theta = \frac{3\pi}{4} $

So,

$$ \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} $$

and

$$ \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} $$

The coordinates are:

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