Math

What is a unit circle in trigonometry, and how is it used to define the trigonometric functions?

A unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. The equation of the unit circle is given by:

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

In trigonometry, the unit circle is used to define the trigonometric functions sine and cosine. For a given angle $ \theta $, measured from the positive x-axis, the coordinates of the corresponding point on the unit circle are $(\cos(\theta), \sin(\theta))$. These definitions can be extended to all real numbers by considering the angle $ \theta $ to be the result of wrapping the real line around the unit circle.

Furthermore, the unit circle allows us to define the other trigonometric functions as follows:

• Tangent: $$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $$
• Cosecant: $$ \csc(\theta) = \frac{1}{\sin(\theta)} $$
• Secant: $$ \sec(\theta) = \frac{1}{\cos(\theta)} $$
• Cotangent: $$ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} $$

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 $\frac{3\pi}{4}$ on the unit circle

To find the coordinates of $$ \frac{3\pi}{4} $$ on the unit circle, we use the unit circle properties:

The x-coordinate is:

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

The y-coordinate is:

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

So, the coordinates are $$ \left( -\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $$

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:

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