Unit Circle

Determine the cosine of an angle on the unit circle at 45 degrees

To find the cosine of an angle at $45^{\circ}$ on the unit circle, recall that:

$$ \cos(45^{\circ}) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} $$

Find the exact values of tan(theta) for theta on the unit circle at each 30-degree increment, and explain the symmetry properties of the tangent function on the unit circle

For each 30-degree increment ($ \theta $) on the unit circle, we have:

$ \tan(0^\circ) = 0 $

$ \tan(30^\circ) = \frac{1}{\sqrt{3}} $

$ \tan(60^\circ) = \sqrt{3} $

$ \tan(90^\circ) = \text{undefined} $

$ \tan(120^\circ) = -\sqrt{3} $

$ \tan(150^\circ) = -\frac{1}{\sqrt{3}} $

$ \tan(180^\circ) = 0 $

$ \tan(210^\circ) = \frac{1}{\sqrt{3}} $

$ \tan(240^\circ) = \sqrt{3} $

$ \tan(270^\circ) = \text{undefined} $

$ \tan(300^\circ) = -\sqrt{3} $

$ \tan(330^\circ) = -\frac{1}{\sqrt{3}} $

$ \tan(360^\circ) = 0 $

The tangent function is periodic with a period of $ 180^\circ $, hence $ \tan(\theta + 180^\circ) = \tan(\theta) $.

Find the exact values of trigonometric functions for given unit circle angles

Given the angle $ \theta = \frac{5\pi}{4} $, find the exact values of $ \sin(\theta) $, $ \cos(\theta) $, and $ \tan(\theta) $:

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

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

$$ \tan\left(\frac{5\pi}{4}\right) = 1 $$

Find the value of tan at π/4 on the unit circle

To find the value of $ \tan(\frac{\pi}{4}) $ on the unit circle, we use the definition of tangent, which is the ratio of sine to cosine:

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

At $ \theta = \frac{\pi}{4} $, both $ \sin(\frac{\pi}{4}) $ and $ \cos(\frac{\pi}{4}) $ are equal to $ \frac{\sqrt{2}}{2} $:

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

Find the angles at which sin(θ) = cos(θ)

To find the angles where $ \sin(\theta) = \cos(\theta) $, we know that:

$$ \sin(\theta) = \cos(\theta) $$

Dividing both sides by $ \cos(\theta) $, we get:

$$ \tan(\theta) = 1 $$

Thus, $ \theta $ must be an angle where the tangent is 1. We know that $ \tan(\theta) = 1 $ at:

$$ \theta = \frac{\pi}{4} + n\pi $$

where $ n $ is any integer. So, the angles are:

$$ \theta = \frac{\pi}{4}, \frac{5\pi}{4}, \frac{9\pi}{4}, … $$

Determine the sine value at an angle of π/4 on the unit circle

To determine the sine value at an angle of $ \frac{\pi}{4} $ on the unit circle, recall that at this angle, the coordinates are:

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

The sine value corresponds to the y-coordinate:

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

Determine tan(θ) from the unit circle at point P(x,y)

To determine $ \tan(\theta) $ from the unit circle at point $ P(x,y) $, recall that

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

On the unit circle, you have $ P(x,y) = (\cos(\theta), \sin(\theta)) $, so

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

Ensure that $ x \neq 0 $ to avoid division by zero.

Create a colorful circle pattern using points on the unit circle with $cos(\theta)$ and $sin(\theta)$

To create a colorful circle pattern, you can use points on the unit circle defined by $\cos(\theta)$ and $\sin(\theta)$ where $0 \leq \theta \leq 2\pi$. Each point coordinates can be calculated as:

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

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

For instance, if you plot points for $\theta$ in multiples of $\frac{\pi}{6}$, you will get 12 equally spaced points around a circle.

Determine the coordinates on the unit circle for the angle -2/3π

To determine the coordinates on the unit circle for the angle $-\frac{2}{3}π$, we first convert this angle to its corresponding positive angle by adding $2π$:

$$ -\frac{2}{3}π + 2π = \frac{4π}{3} $$

Now, we find the coordinates corresponding to the angle $\frac{4π}{3}$ on the unit circle. This angle is in the third quadrant, where both sine and cosine are negative:

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

Find the value of arcsin(1/2)

To find the value of $ \arcsin(\frac{1}{2}) $, we need to determine the angle $ \theta $ whose sine is $ \frac{1}{2} $.

From the unit circle, we know:

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

The angle $ \theta $ that satisfies this in the range $ [-\frac{\pi}{2}, \frac{\pi}{2}] $ is:

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

Thus, $ \arcsin(\frac{1}{2}) = \frac{\pi}{6} $.