Unit Circle

Methods to Quickly Memorize the Unit Circle

$$Method\ 1: \ Use\ Symmetry$$

$$Explanation: \ The\ unit\ circle\ is\ symmetric\ about\ the\ x-axis,\ y-axis,\ and\ the\ origin.\ By\ knowing\ the\ key\ points\ in\ the\ first\ quadrant,\ you\ can\ easily\ deduce\ the\ corresponding\ points\ in\ the\ other\ three\ quadrants.\ Specifically,\ remember\ coordinates\ of\ (\frac{\pi}{6},\ \frac{\sqrt{3}}{2},\ \frac{1}{2})\ and\ (\frac{\pi}{4},\ \frac{\sqrt{2}}{2},\ \frac{\sqrt{2}}{2})\ and\ (\frac{\pi}{3},\ \frac{1}{2},\ \frac{\sqrt{3}}{2}) \ for\ first\ quadrant.\ Rest\ will\ be\ just\ reflections. $$

Find the cosine and sine of pi/10 on the unit circle

To find the cosine and sine of $ \frac{\pi}{10} $ on the unit circle, we use the following steps.

Since $ \frac{\pi}{10} $ is an angle in radians, we can find its coordinates on the unit circle. The coordinates of an angle $ \theta $ on the unit circle are given by $ (\cos(\theta), \sin(\theta)) $.

Therefore, for $ \theta = \frac{\pi}{10} $:

$$ \cos(\frac{\pi}{10}) $$ $$ \cos(\frac{\pi}{10}) \approx 0.9511 $$

$$ \sin(\frac{\pi}{10}) $$ $$ \sin(\frac{\pi}{10}) \approx 0.3090 $$

Thus, the cosine and sine of $ \frac{\pi}{10} $ on the unit circle are approximately $ 0.9511 $ and $ 0.3090 $, respectively.

Find the sine, cosine, and tangent values for 45 degrees on the unit circle

To find the sine, cosine, and tangent values for $45^\circ$ on the unit circle, we use the following formulas:

$$\sin 45^\circ = \frac{1}{\sqrt{2}}$$

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

$$\tan 45^\circ = 1$$

Therefore, the sine, cosine, and tangent values for $45^\circ$ are $\frac{1}{\sqrt{2}}$, $\frac{1}{\sqrt{2}}$, and $1$ respectively.

Learning the Unit Circle Easily

First, understand the basics of the unit circle. The unit circle is a circle with a radius of 1, centered at the origin (0, 0) of a coordinate plane.

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

Next, memorize key angles and their coordinates in both degrees and radians. For example:

$$0^{\circ} (0, 1)$$

$$90^{\circ} \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)$$

$$180^{\circ} (0, -1)$$

$$270^{\circ} \left( -\frac{1}{2}, \frac{-\sqrt{3}}{2} \right)$$

Use symmetry to find coordinates of other angles.

Find the value of tan θ given that θ is an angle on the unit circle with a terminal side passing through the point (-1/2, -√3/2)

To find the value of $$\tan \theta $$, we use the fact that tan is defined as the ratio of the y-coordinate to the x-coordinate on the unit circle.

Given the point $$\left( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \right)$$, we have:

$$\tan \theta = \frac{y}{x} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}}$$

Simplify the expression:

$$\tan \theta = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{1} = \sqrt{3}$$

Thus, the value of $$\tan \theta$$ is $$\sqrt{3}$$.

Calculate the cosine and sine of the angle pi/3 using the unit circle

Using the unit circle, we know that the angle $\frac{\pi}{3}$ corresponds to 60 degrees.

From the unit circle properties:

The coordinates at $\frac{\pi}{3}$ are $(\frac{1}{2}, \frac{\sqrt{3}}{2})$.

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

Answer: Cosine: $\frac{1}{2}$, Sine: $\frac{\sqrt{3}}{2}$.

What is the value of sin(30°) + cos(60°) + tan(45°) on the unit circle?

To solve for $\sin(30°) + \cos(60°) + \tan(45°)$, we need to find the individual values:

$$ \sin(30°) = \frac{1}{2} $$

$$ \cos(60°) = \frac{1}{2} $$

$$ \tan(45°) = 1 $$

Adding these values together:

$$ \sin(30°) + \cos(60°) + \tan(45°) = \frac{1}{2} + \frac{1}{2} + 1 = 2 $$

Therefore, the value is 2.

Find the coordinates of the point where the angle 5π/4 radians intersects the unit circle

First, we need to convert the angle $\frac{5\pi}{4}$ radians into degrees. We know that:

$$ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} $$

Thus,

$$ \frac{5\pi}{4} \text{ radians} = \frac{5\pi}{4} \times \frac{180}{\pi} \text{ degrees} = 225 \text{ degrees} $$

On the unit circle, the coordinates corresponding to an angle of $225^{\circ}$ (or $\frac{5\pi}{4}$ radians) can be found using the cosine and sine functions:

$$ \cos(225^{\circ}) = -\frac{\sqrt{2}}{2} $$

$$ \sin(225^{\circ}) = -\frac{\sqrt{2}}{2} $$

Therefore, the coordinates are:

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

Locate -π/2 on a Unit Circle

To locate $-\pi/2$ on the unit circle, we can follow these steps:

1. Start at the positive x-axis (0 radians).

2. Move clockwise because the angle is negative.

3. Since $-\pi/2$ radians equals -90 degrees, move 90 degrees clockwise from the positive x-axis.

4. This will place you on the negative y-axis.

Therefore, the coordinates for $-\pi/2$ on the unit circle are (0, -1).

Find the Trigonometric Values Using the Unit Circle

Given the angle $\theta = \frac{2\pi}{3}$, find the values of $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$ using the unit circle on a GDC TI calculator.

1. Locate the angle $\theta = \frac{2\pi}{3}$ on the unit circle.

2. The coordinates of the point where the terminal side intersects the unit circle are $(-\frac{1}{2}, \frac{\sqrt{3}}{2})$.

3. Hence, $\sin(\theta) = \frac{\sqrt{3}}{2}$, $\cos(\theta) = -\frac{1}{2}$, and $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\sqrt{3}}{2} \div -\frac{1}{2} = -\sqrt{3}$.

Therefore, the trigonometric values are:

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

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

$$\tan(\frac{2\pi}{3}) = -\sqrt{3}$$