February 09, 2011 - Degree and Median Measure

Hello everyone! As I am typing this blog entry I am finishing up out pre-calculus homework, reviewing our notes as well as beginning to finish other homework. This will be a recap of today's class (February 09, 2011) as well as this will be the first memorable blog post to start off our blog!

Note: When you see something like this: #/#, the "/" signifies a fraction. So if you see 5/5 it means 5 over 5 and not 5 divided by 5.

So what did we learn today? We learned many things but to simplify it more, we have a new formula/equation that we must know for the provincial pre-calculus exam. What is this formula/equation?

S=rθ

Respectively, S would stand for the arc length, r would stand for the radius and theta would stand for the measured angle in radians and not degrees. Do you guys remember there are two ways of measuring angles? We have been taught to use degrees but we can also use radians.

Now there were important revolution into radian measurements that we should memorize and know by heart for the pre-calculus exam.

1. 1 revolution = 2π radians (same as 360°)
2. 1/2 revolution = π radians (same as 180°)
3. 1/4 revolution = π/2 radians (same as 90°)
4. 1/8 revolution = π/4 radians (same as 45°)
5. 1/6 revolution = π/3 radians (same as 60°)
6. 1/12 revolution = π/6 radians (same as 30°)

Keep in mind that a revolution is how many times you go around the circle.
So 1 full revolution means going through a complete 360° of the circle.

We then moved onto the conversion from degrees to radians and radians to degrees. It's fairly easy and all you have to do is remember these two very easy formulas/equations.

To get from degrees to radians we will divide by 180.
(The # stands for whatever number of degree you are working with. If it's 8 then you will put 8°)

#°=π/180 (radians)

Example: You are given 60° and are asked to find the radian of 60°. Let us go step by step on how this will work.

Step 1: We will first multiply the given degree with π/180 (radians) Note: I added radians in a parentheses because we don't necessarily have to write it down. Since our given degree is 60°, we will set it up so that it is multiplying with π/180.

Step 2: After setting up our equation we will cross multiply or simplify. Keep in mind because 60° isn't seen as a fraction, there is still a "1" as its denominator.




Step 3:



How did we get to there? Simply if we ask ourselves what's the largest number that can be divided into both 60 and 180? 60! So we divided 60 by 60 to get 10 and 180 by 60 to get 30. This can be simplified even further by dividing by 10 or just taking the zeros away (make sure that if you take out a zero, you take out the same amounts. If you take 2 from the bottom, you take 2 from the top and vice versa)




In the end that is the radian from the measurement of 60°. You can make it even simpler where at step 2, when you simplify, you could have taken the zeros away to be left with 6 and 18 then simplify straight to the final answer you see above.

To get from radians to degrees we will divide by π.

Example: You are given π/3 = 180°/π. Convert that to degrees.

Step 1: We will simplify first than cross multiplying. (You may cross multiply but that will give you more work later. I recommend simplifying. Both 180° and 3 can be simplified because the largest number that can be divided into both are 3. So by dividing 180° by 3 will give you 60 and 3 by 3 will give you 1.

Step 2: Now we can cross multiply.






Step 3: As you can see the answer is 60π/3 radians WHICH in degrees is 60

That will wrap up the two formulas/equations we have learned today. Now here are some quick reminders and tips:

1. When working with S=rθ, we always work in radians and never degrees.
   
   
2. When an angle is shown going clockwise, it is a negative angle. When an angle is shown going counterclockwise, it is a positive angle.
   
   
3. There are 4 quadrants. Remember the CAST rule. Cosine is positive in quadrant 4, all (cosine, sine and tangent) are positive in quadrant 1, sine is positive in quadrant 2 and tangent is positive in quadrant 3.

Don't forget we have an assignment due tomorrow.
It's "Degree and Median Measure" questions 1 - 24