Unit Circle

Find the sine value for 5π/6 on the unit circle

To find the sine value for $ \frac{5\pi}{6} $ on the unit circle, we follow these steps:

First, understand that $ \frac{5\pi}{6} $ is in the second quadrant.

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

In the second quadrant, sine is positive, and we know:

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

Therefore,

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

Find the value of sine and cosine for specific angles using the unit circle

Using the unit circle, find the values of $\sin$ and $\cos$ for the angle $\frac{\pi}{4}$.

For $\theta = \frac{\pi}{4}$:

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

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

Identify the angle on the unit circle for cos(θ) = 1/2

To identify the angles where $\cos(\theta) = \frac{1}{2}$, we look at the unit circle:

In the first quadrant, $\theta = \frac{\pi}{3}$, and in the fourth quadrant, $\theta = \frac{5\pi}{3}$.

Determine the cosine value of an angle given on the unit circle

Given an angle $ \theta $ on the unit circle, we need to determine the value of $ \cos(\theta) $.

For example, if $ \theta = \frac{\pi}{3} $, we can use the unit circle to find:

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

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

Find the coordinates of a point on the unit circle corresponding to an angle of 5π/6

To find the coordinates of a point on the unit circle at an angle of $ \frac{5\pi}{6} $, we use the unit circle definitions for sine and cosine:

$$ \text{cos}(\theta) = \text{x-coordinate} $$

$$ \text{sin}(\theta) = \text{y-coordinate} $$

For $ \frac{5\pi}{6} $:

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

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

So, the coordinates are:

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

Describe the unit circle and determine the coordinates of a point with a given angle

The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane, i.e., at (0, 0). The equation of the unit circle is:

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

Given an angle $\theta$ measured in radians from the positive x-axis, the coordinates $(x, y)$ of the corresponding point on the unit circle can be determined using trigonometric functions:

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

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

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

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

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

So, the coordinates of the point are:

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

Find the solutions to arcsin(x) = π/6 using the unit circle

To find the solutions for $ \arcsin(x) = \frac{\pi}{6} $ using the unit circle, we need to identify the values of $x$ for which the angle is $ \frac{\pi}{6} $:

• On the unit circle, $ \arcsin(x) = \frac{\pi}{6} $ corresponds to the $y$-coordinate of the point where the angle from the positive $x$-axis is $ \frac{\pi}{6} $.
• At $ \frac{\pi}{6} $, the coordinates are $(\frac{\sqrt{3}}{2}, \frac{1}{2})$.
• Thus, $x = \frac{1}{2}$.

Therefore, the solution is:

$$ x = \frac{1}{2} $$

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

To find the values of $ \tan(\theta) $ for specific angles on the unit circle, consider the angles $ \theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} $:

For $ \theta = \frac{\pi}{4} $:

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

For $ \theta = \frac{3\pi}{4} $:

$$ \tan\left(\frac{3\pi}{4}\right) = -1 $$

For $ \theta = \frac{5\pi}{4} $:

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

For $ \theta = \frac{7\pi}{4} $:

$$ \tan\left(\frac{7\pi}{4}\right) = -1 $$

Find the coordinates on the unit circle corresponding to an angle of θ

To find the coordinates on the unit circle for an angle $ \theta $, we use the trigonometric functions sine and cosine. The coordinates are given by:

\n

$$ (x, y) = (\cos(\theta), \sin(\theta)) $$

\n

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

\n

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

\n

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

\n

Thus, the coordinates are:

\n

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

Find the points on the unit circle where the secant of the angle is equal to 2, and prove their coordinates

To find points on the unit circle where $ \sec(\theta) = 2 $, recall that:

$$ \sec(\theta) = \frac{1}{\cos(\theta)} $$

Thus, we need:

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

So:

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

The angles on the unit circle with $ \cos(\theta) = \frac{1}{2} $ are:

$$ \theta = \frac{\pi}{3} \text{ and } \theta = \frac{5\pi}{3} $$

The corresponding points on the unit circle are:

For $ \theta = \frac{\pi}{3} $:

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

For $ \theta = \frac{5\pi}{3} $:

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