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Math 103: Precalculus12 chapters | 94 lessons | 10 flashcard sets

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Lesson Transcript

Instructor:
*Tyler Cantway*

Tyler has tutored math at two universities and has a master's degree in engineering.

Memorizing the unit circle can be a daunting task, but this lesson will show you a pattern to help you memorize the points, degree measures, and radian measures for the entire first quadrant.

Some of the more useful study skills you can learn are actually memorization tricks to make complicated information simpler. When it comes to math, sure, there are things you just have to remember. These can be things like basic addition and subtraction all the way up to trig functions.

But there are some things in math where it's more helpful to remember a few short patterns rather than try to memorize everything. For trigonometric calculations, the most important thing you can memorize is the **unit circle**. The unit circle contains several useful angles along with their *x* and *y* coordinates.

Memorizing the unit circle can be extremely helpful because, if you have it memorized, you can calculate the trig values of the angles without a calculator. You can even go backwards by taking a trig value and finding the angle that created it. Now, you may be worried that memorizing this will be impossible. Don't worry. There are a few patterns that we can learn to make it a lot easier and more useful.

Even though the unit circle has four quadrants, most of them are just small twists on the first quadrant. Believe it or not, there are a few patterns in the first quadrant to help us memorize it.

The first quadrant angles range from zero degrees to 90 degrees. To get angles in between, we split it in half and in thirds. This gives us angles of zero, 30, 45, 60, and 90 degrees. Now that we know the degree measures for the angles, we need to write the radian measures. You can convert all the degree measures into radian measures by multiplying them by 2*pi*/360.

Let's do this for zero and 90 degrees. Zero multiplied by anything is zero, so that angle is zero radians. 90*2*pi*/360 simplifies to *pi*/2 radians. For the other three angles, you can convert or you can use a pattern. Take the first number from the angle in degrees and put *pi* over them. Reverse the order and you have *pi*/6, *pi*/4, and *pi*/3.

Knowing the angles is the first step, but to be able to do calculations, we have to know the *x* and *y* coordinates for each angle. Again, we will start with the zero and 90 degree angles. These are straightforward. The coordinates are read directly off the graph. The point for a zero angle on the unit circle is the point **(1,0)**. The point for a 90 degree or *pi*/2 radian angle is the point **(0,1)**.

Now we need to come up with the *x* and *y* coordinates for the three angles in between. These will all be fractions, so our first step is to put fraction bars in the *x* and *y* places for all the points. The good news is these fractions all follow a pattern.

The bottom number for all the fractions is 2. The top number for all the fractions will be under a square root. The rest is as easy as one, two, three. The *x* coordinates from the top are **1**, **2**, **3**. The *y* coordinates from the bottom are also **1**, **2**, and **3**. We can simplify the square root of 1 to just 1, and we are finished!

We wouldn't just memorize this for nothing. Each of these values will help us with calculations of trig values. To find the **sine (sin)** of any angle on the unit circle, we take its * y* value. To find the

This works for degrees and radians. It also works in reverse. We can be given a sine, cosine, or tangent value and know the angle that created it.

The first quadrant of the unit circle is the most important because it is the only section of the unit circle where you can use angles to find trig values and vice versa for ALL the functions. Memorizing this looks intimidating. But when we notice the pattern it's easy to find out how it all relates. We divide it into halves and thirds to get our important angles, and then we find their coordinate values by using square roots in the top, the number two in the bottom and then using 1, 2, 3.

After viewing this video and with some practice, students should be able to memorize the first quadrant to solve trigonometric calculations.

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Math 103: Precalculus12 chapters | 94 lessons | 10 flashcard sets

- Go to Functions

- Graphing Sine and Cosine 7:50
- Graphing Sine and Cosine Transformations 8:39
- Graphing the Tangent Function: Amplitude, Period, Phase Shift & Vertical Shift 9:42
- Unit Circle: Memorizing the First Quadrant 5:15
- Special Right Triangles: Types and Properties 6:12
- Law of Sines: Definition and Application 6:04
- Law of Cosines: Definition and Application 8:16
- The Double Angle Formula 9:44
- Converting Between Radians and Degrees 7:15
- How to Solve Trigonometric Equations for X 4:57
- List of the Basic Trig Identities 7:11
- Go to Trigonometry

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