Question

The Unit Circle Tutorial
Creator of Tutorial: BloomLocke367

Answer 1

This first two things you must understand before delving into \(\large\color{navy}{\mathbb{{The~Unit~Circle}}}\), is the \(\large\mathbb{\color{red}{Pythagorean~ Theorem}}\) and the use of \(\large\mathbb{\color{hotpink}{Sine}}\), \(\large\mathbb{\color{lime}{Cosine}}\), and \(\large\mathbb{\color{teal}{Tangent}}\). \(\large\bf\color{cornflowerblue}{(SohCahToa)}\)

Answer 2

\(\huge\mathbb{\color{red}{The~Pythagorean~Theorem:}}\)

If you are familiar with trigonometry and right triangles, this should be easy for you.

When you have a \(\bf\color{#FF4000}{right ~triangle}\), a triangle with a \(\bf\color{#FF4000}{90° angle}\), there are two legs and a hypotenuse (the longest side to the triangle).

We'll call the first leg \(\color{green}{a}\), the second leg \(\color{orange}{b}\), and the hypotenuse \(\color{blue}{c}\).

To solve for a side-length, you use the \(\large\mathbb{\color{red}{Pythagorean~ Theorem}}\), which is:
\(\huge \color{green}{a}^2\LARGE +\huge\color{orange}{b}^2\LARGE =\huge\color{blue}{c}^2\)

All you have to do is plug in your given values and solve for the missing variable.

Answer 3

\(\bf\huge\color{cornflowerblue}{SOHCAHTOA:}\)

\(\large\mathbb{\color{hotpink}{Sine}}\theta =\color{hotpink}{\frac{\color{hotpink}{opposite}}{hypotenuse}}\)
\(\large\mathbb{\color{lime}{Cosine}}\theta =\color{lime}{\frac{adjacent}{hypotenuse}}\)
\(\large\mathbb{\color{teal}{Tangent}}\theta =\color{teal} {\frac{opposite}{adjacent}}\)

The \(\color{blue}{\underline{ hypotenuse}}\) is the longest side.
The \(\color{red}{\underline{ adjacent}}\) side is the side length that is next to the given angle (\(\theta\)).
The \(\color{green}{\underline{opposite}}\) side is the side length that is across from the given angle (\(\theta\)).

Answer 4

It is also important that you know how to measure your angles in radians, and how to convert from degrees to radians, and vice versa.

\(\large 360°=2\pi~ radians\)
\(~~~~~~~or\)
\(\large 180°=\pi~radians\)

Answer 5

To convert from degrees to radians, you must take the angle in degrees and multiply it by \(\Large\frac{\pi}{180}\).

\(\Large\bf Radians=\theta\times \LARGE \frac{\pi}{180}\)

For example, if \(\Large\theta=180°\),
you plug 180 in for theta.

\(\LARGE\bf 180°\times\frac{\pi}{180}=\pi~radians\)

Answer 6

Now, we can move onto the \(\Large\mathbb{\color{navy}{Unit~Circle}}\).
It is called the unit circle because it has a radius of 1 and is centered at the origin (0,0). It also follows the equation \(\large x^2+y^2=1\)


let r=1 in this, as we already know the radius is 1.

Given what I told you earlier about \(\bf\color{cornflowerblue}{SOHCAHTOA}\), you know that \(\bf\large\color{hotpink}{sin}\theta=\color{hotpink}{\frac{opposite}{hypotenuse}}\) and \(\bf\large\color{lime}{cos}\theta=\color{lime}{\frac{adjacent}{hypotenuse}}\) .

Given this, you know that
\(\bf\large\color{hotpink}{sin}\theta=\color{hotpink}{\frac{y}{1}}=\color{hotpink}{y}\)
and
\(\bf\large\color{lime}{cos}\theta=\color{lime}{\frac{x}{1}}=\color{lime}{x}\)

Answer 7

From this, you can construct the unit circle.
The unit circle has key points at 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, and 360°--as shown below



\(\bf **Notice: ~There ~is~ a~ pattern ~with ~the~ angle~ measures.\\ Add~ 30,~ 15, ~15, ~30~ and~ repeat**\)

You can use what I said earlier, and convert the angles to radians. You can also use Sine and Cosine to compute the find the coordinates on the circle at those angles.

Answer 8

After you do that, you will get this--the Unit Circle:


Remember:
\(\bf\color{hotpink}{sin\theta=y}\) and \(\bf\color{lime}{cos\theta=x}\).. You can check this by using the coordinates on the circle.

Answer 9

It is also helpful to know that \(\bf\color{teal}{tangent=\Large \frac{\color{hotpink}{sin}}{\color{lime}{cos}}\normalsize=\Large \frac{\color{hotpink}{y}}{\color{lime}{x}}}\).

That's really all there is to the unit circle! I may add a part 2 with more advanced information in the future.