## Sunday, April 10, 2011

### Friday, April 8 2011 Post

On Friday, we checked over homework, then started on Day 2 of Unit Circle and Radians. You can find homework answers on moodle.

10-4 (Day 2) More Unit Circle and Radians.
Overall, we learned in this section how to:
2. fill in our very first unit circle!!

First off, we defined what a radian actually is: the angle created by placing a radius along the circumference of a circle. (If this doesn't make sense, think of a slice of pizza cut in triangular shapes, from the inside out. Take the crust, and that is what the radian is.)

VERY IMPORTANT: for this chapter, and possibly beyond this, you need to switch your calculator mode. Here's how it's done:
1. press home then #5
2. choose #2 for settings
3. choose #1 for general
4. tab down until you reach Angle
6. tab down until you reach Make Default
7. press enter
8. select OK

If you forget to switch modes, you'll notice that you're not going to get your answers in either degree or radian mode.

Conversions
If we know that 360 degrees around a circle gives us an arc length of 2π, then 180 degrees around will give us half of that..which is just π.
One fourth of that will gives us π divided by two so it's just π/2.
This information will help us convert between degree and radian, and radian and degree.

$\mbox{deg} = \mbox{rad} \cdot \frac {180^\circ} {\pi}$

$\mbox{rad} = \mbox{deg} \cdot \frac {\pi} {180^\circ}$
Think of it like this. What you want to get rid of it on the bottom, and what you want is on the top.
We then practiced converting some measures.
$1 \mbox{ rad} = 1 \cdot \frac {180^\circ} {\pi} \approx 57.2958^\circ$

1 deg. = 1xπ/180 = π/80 (if you're converting to radians, leave in simplest form, so leave the π in)

Unit Circle
It's going to be a bit difficult to explain, but once it's actually filled out, the unit circle does make sense.
Let's start off with a 30-60-90 triangle.
The measures are shown, and the way we use this information is by finding
the cos and sin of some angles.
cos(60)=1/2
sin(60)=√3/2
cos(30)=√3/2
sin(30)=1/2

Now, for a 45-45-90 triangle, We will do the same, but because two angles are the same (45) we don't need to do double the work.
cos(45)=√2/2 (remember, we have to multiply by √2 to get the radial out from under and put it on top)
sin(45)=√2/2

Now that we have these angles, we can use them to help apply them to the unit circle.

This is a completed Unit Circle. It's complicated, but once taken apart, it's essentially just one piece of information being copied four times over. Let's break it down:

FINDING OUR POINTS

1. Let's start with the points. On quadrant I, we have two points that are easy to remember.

At 0 degrees, we have (1,0).

At 90 degrees, we have (0,1).

These could be the first things that you put down to help you remember the unit circle, but first, a trick.

All the angles on the unit circle have a pattern(at least the ones we need to know).

The pattern is up 30 degrees, up 15, up another 15, then up 30. It keeps going around and around.

30, 15, 15, 30 and so on. This should help identify where points are when they're not exactly on the axis.

FINDING OUR DEGREES

2. After you have all your degrees filled in, refer back to the triangles. We took the sin and cos of some angles, and now we can use them here.

FOR ALL 45 degree angles (45, 135, 225, and 315) the sin/cos points will be the same because a 45-45-90 triangle only really has that one measure-- √2/2. The positive and negative signs will change because of which quadrant the points will be in, but the numbers are the same.

FOR ALL 30-60 angles, we can use what we found from our triangle above. If you notice on the picture of quad. 1, at 30 degrees, it's coordinates are (√3/2, 1/2). These points are the cos and sin of 30 degrees from our triangle. If you look at 60 degrees, it's also the cos and sin from the triangle. This is how they connect!!!

The same idea from the 45-45 triangle applies to the 30-60 triangle, except you take your points and SLIDE them over. DO NOT FLIP!!!! SLIDE!!!!

3. Finding the radian points are a bit more difficult, but once a simple concept is understood, it becomes less difficult. Just think of each point and a fraction of the whole unit circle. Let's start with what we know--

all the way around, 360 degrees, is equivalent to 2π. From there, we can find the other points. So 90, which is one fourth of 360, or one half of 180 (just π), will be π/2.

Now that we have our two axis points from quad. 1, we can find the ones on the circle.

Take 45 degrees. It's one half of 90 degrees, so take the radian from 90 (which is π/2) and divide that

by two to get the measure for 45.

That is one way of thinking of it.

Another way is by thinking of the top two quadrants. Because 180 is just π, then you can think of 90 and one half of that,

making it π/2.

Then that means that 60 degrees is one third of π, making it π/3.

If that is true, which it is, then 30 is one sixth of π, so π/6.

This process of thinking and each radian point as a fraction can help simplify the process.

If this still doesn't make sense, here's an extra video that helped me. Mr. Cope can also help (considering he is our teacher..)

and the TLC has great tutors!