Math

Find the tangent of the angle \( \theta \) in the unit circle

Consider the unit circle, where the radius is 1. Let $ \theta $ be an angle in standard position.

The coordinates of the point on the unit circle at an angle $ \theta $ are $(\cos \theta, \sin \theta)$.

The tangent of the angle $ \theta $ is given by

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

For example, if $ \theta = 45^\circ $, then $ \sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2} $.

Thus, $$ \tan 45^\circ = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 $$

Evaluate the cosine of an angle using the unit circle in the complex plane

$$ \text{Given an angle } \theta \text{, we need to find } \cos(\theta) \text{ using the unit circle in the complex plane.} $$

$$ \text{On the unit circle, the coordinates of a point } P \text{ corresponding to the angle } \theta \text{ are } (\cos(\theta), \sin(\theta)). $$

$$ \text{Thus, } \cos(\theta) \text{ is simply the x-coordinate.} $$

$$ \text{For example, if } \theta = \frac{\pi}{3}, \text{ the coordinates on the unit circle are } (\cos(\frac{\pi}{3}), \sin(\frac{\pi}{3})). $$

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

Find the value of sec(π/3) on the unit circle

To find the value of $\sec(\frac{\pi}{3})$, we need to first determine the cosine of $\frac{\pi}{3}$.

On the unit circle, $\cos(\frac{\pi}{3}) = \frac{1}{2}$.

The secant function is the reciprocal of the cosine function, so

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

Find the equation of a circle with center at (h, k) and radius 1

The general equation of a circle with center $(h, k)$ and radius $r$ is given by:

$$ (x – h)^2 + (y – k)^2 = r^2 $$

For a unit circle, the radius $r = 1$. Therefore, the equation becomes:

$$ (x – h)^2 + (y – k)^2 = 1 $$

Find the value of sin(π/3) using the unit circle

$$\text{The angle } \frac{\pi}{3} \text{ is equivalent to } 60^{\circ}.$$

$$\text{On the unit circle, the coordinates for } 60^{\circ} \text{ are } \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right).$$

$$\sin(\frac{\pi}{3}) \text{ is the y-coordinate, which is } \frac{\sqrt{3}}{2}.$$

Find the sine and cosine of 𝜋/6 radians on the unit circle

To find the sine and cosine of $\frac{\pi}{6}$ radians on the unit circle, we need to recall the standard angle values:

At $\frac{\pi}{6}$ radians:

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

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

Thus, the cosine of $\frac{\pi}{6}$ radians is $\frac{\sqrt{3}}{2}$, and the sine of $\frac{\pi}{6}$ radians is $\frac{1}{2}$.

Find the coordinates and trigonometric values for an angle on the unit circle

Consider an angle $ \theta = \frac{7\pi}{6} $ on the unit circle. We need to find the coordinates of the point on the unit circle corresponding to this angle, as well as the sine and cosine values.

First, identify the reference angle: $$ \theta_{ref} = \pi – \frac{7\pi}{6} = \frac{\pi}{6} $$

Next, find the coordinates for the reference angle $ \frac{\pi}{6} $:

$$ \left( \cos\left(\frac{\pi}{6}\right), \sin\left(\frac{\pi}{6}\right) \right) = \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) $$

Since $ \theta = \frac{7\pi}{6} $ is in the third quadrant, both sine and cosine are negative:

$$ \left( \cos\left(\frac{7\pi}{6}\right), \sin\left(\frac{7\pi}{6}\right) \right) = \left( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \right) $$

How to Remember the Unit Circle Fast

$$\text{To remember the unit circle, focus on key angles and their coordinates. Start with } 0^\circ, 30^\circ, 45^\circ, 60^\circ, \text{ and } 90^\circ.$$

$$\text{For example, at } 0^\circ, \text{ the coordinates are } (1, 0).$$

$$\text{At } 30^\circ, \text{ the coordinates are } \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right).$$

$$\text{At } 45^\circ, \text{ the coordinates are } \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right).$$

$$\text{At } 60^\circ, \text{ the coordinates are } \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right).$$

$$\text{At } 90^\circ, \text{ the coordinates are } (0, 1).$$

$$\text{Memorize these points, and use symmetry to fill in the rest of the circle.}$$