Unit Circle

Find the values of tan(θ) for θ in the unit circle at 0, π/4, π/3, and π/2

To determine the values of $ \tan(\theta) $ for $ \theta $ in the unit circle at $ 0 $, $ \frac{\pi}{4} $, $ \frac{\pi}{3} $, and $ \frac{\pi}{2} $, we evaluate the tangent function at these angles:

For $ \theta = 0 $:

$$ \tan(0) = 0 $$

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

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

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

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

For $ \theta = \frac{\pi}{2} $:

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

Find the exact value of sin(π/4) on the unit circle

To find the exact value of $ \sin(\frac{\pi}{4}) $ on the unit circle, we recognize that $ \frac{\pi}{4} $ is equivalent to $ 45^{\circ} $.

On the unit circle, the coordinates for $ \frac{\pi}{4} $ are $ \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $.

The sine value is the y-coordinate, so:

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

Explain the concept of a unit circle, including its importance in trigonometry and how it relates to the coordinates of points on the circle

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

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

The unit circle is fundamental in trigonometry as it defines the sine and cosine functions for all real numbers. For any angle $\theta$, the coordinates of the corresponding point on the unit circle are $(\cos(\theta), \sin(\theta))$. These coordinates are derived from the definitions:

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

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

Additionally, the unit circle helps in visualizing and understanding periodic properties of trigonometric functions and their symmetries.

Find the angle in degrees corresponding to 7π/6 radians on the unit circle

To convert $\frac{7\pi}{6}$ radians to degrees, we use the conversion factor:

$$ 180^{\circ} = \pi \text{ radians} $$

Thus,

$$ \frac{7\pi}{6} \times \frac{180^{\circ}}{\pi} = 210^{\circ} $$

The angle in degrees is:

$$ 210^{\circ} $$

Find the coordinates on the unit circle where the tangent of the angle is 1

To find the coordinates on the unit circle where $ \tan(\theta) = 1 $, we need to determine the angles $\theta $ for which this condition holds. We know that:

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

For the tangent to be 1, the sine and cosine must be equal. This occurs at angles:

$$ \theta = \frac {\pi}{4} \text{ and } \theta = \frac {5\pi}{4} $$

Now, we find the coordinates on the unit circle for these angles:

$$ \text{At } \theta = \frac {\pi}{4}, \text{ the coordinates are } \left( \frac {\sqrt {2}}{2}, \frac {\sqrt {2}}{2} \right) $$

$$ \text{At } \theta = \frac {5\pi}{4}, \text{ the coordinates are } \left( -\frac {\sqrt {2}}{2}, -\frac {\sqrt {2}}{2} \right) $$

Thus, the coordinates on the unit circle where $ \tan(\theta) = 1 $ are:

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

and

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

Determine the coordinates of the point where the terminal side of an angle of 5π/3 radians intersects the unit circle, and identify its quadrant

The angle $ \frac{5\pi}{3} $ radians is equivalent to 300 degrees (since $ \frac{5\pi}{3} \times \frac{180}{\pi} = 300 $ degrees).

This angle places the terminal side in the fourth quadrant.

In the fourth quadrant, the coordinates on the unit circle corresponding to an angle of 300 degrees are:

$$ ( \cos(300\degree), \sin(300\degree) ) $$

Since $ \cos(300\degree) = \cos(-60\degree) = \frac{1}{2} $ and $ \sin(300\degree) = \sin(-60\degree) = -\frac{\sqrt{3}}{2} $, the coordinates are:

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

Thus, the terminal side intersects the unit circle at $ \left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right) $ in the fourth quadrant.

Find the values of tan(x) at different positions on the unit circle

To find the values of $ \tan(x) $ at different positions on the unit circle, we use the definition:

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

First, let

Find the sine of a negative angle on the unit circle

On the unit circle, the sine of a negative angle $ \theta $ is given by:

$$ \sin(-\theta) = -\sin(\theta) $$

For example, if $ \theta = 30^{\circ} $, then:

$$ \sin(-30^{\circ}) = -\sin(30^{\circ}) = -\frac{1}{2} $$

Calculate tangent values on the unit circle for specific angles