Degree and Radian Measure 1

Hi! My name is Jamie, to those who don't know me, I am the second person sitting in the row near Mr. P's desk. 

Last Thursday, Mr. P introduced to us the Degree and Radian Measure 1, which he also recap earlier in our afternoon class today. 

NOTE: "CIRCULAR FUNCTIONS" IS AN IMPORTANT UNIT of PRECAL40S and probably the hardest one. (maybe?)


An angle is determined by two rays:

                Initial side - the starting point of the ray 

                Terminal side - the position after rotation

We define an angle of rotation by  rotating a ray about its endpoint, creating a vertex.

Angles can be measure using different units, such as:

1. REVOLUTION
2. ROTATIONS
3. DEGREES
4. RADIANS
5. GRADIANS

NOTE:
     Angle measures without units are considered to be RADIANS.
     Angle measures in degrees MUST show the DEGREE SYMBOL.
     OBTUSE ANGLES measures GREATER than 90°.
    ACUTE ANGLES measures LESS than 90°.

*If the rotation of the ray is in COUNTERCLOCKWISE direction, the angle is considered POSITIVE.
   If the rotation is in a CLOCKWISE direction, the angle is considered NEGATIVE. *

 An angle in a coordinate plane is said to be in standard position if:

• Its VERTEX is at the ORIGIN
• Its INITIAL SIDE is on the positive x-axis (Quadrant one)

A REFERENCE ANGLE is the measure of the angle between the TERMINAL ARM and the x-axis

please excuse my messy notes :))

Radian Measure
Radian Measure is the first method that we're going to learn on how to measure the angles in a circles. 
When the arc of a circle has the same length as the radius of a circle, the measure of the central angle that intercepts the arc is 1 radian. 

To find the arc length of a circle, we will use the formula a=θr
               a = arc length
               r = radius
               θ = angle measured in radians

To convert degrees to radians, we multiply the degrees by π/180°.
To convert radians to degrees, we multiply the radian measure by 180°/π.

Translate revolution to degrees, exact radian measure and approximate radian measure.

Example 1: Draw each angle in standard position. Change each degree measure to radian measure and each radian measure to degree measure. Give answers as both exact and approximate measures (if necessary) to the nearest hundredth of a unit. 

A.) for letter a the given measure is a degree number, so we have to convert it into radian measure by multiplying the degree by π/180°.

                    30°  x  π/180° 
                    30°  x  π/180°6 = π/6
                                        π/6  = 0.523598776 
                                        π/6  = 0.52 

(same method for letter B)

π180π180C.) For letter c. the given measure is -60°. we will also do the same step, 

                   -60°  x  π/180° 
                   -60°  x  π/180°3 = - π/3
                                      - π/3  = - 1.047197551
                                      - π/3  = - 1.05

     but the difference in this example is the sign of the degree angle which shows negative, Therefore, the standard position will rotate in clockwise direction.  

*NO NEGATIVE ANGLES*
e.) In letter E, Mr. P gave example on how to measure without using formula.

g.) we are given a approximate radian measure, so we have to convert it into degree measure by multiplying  180°/π.

               1 radian  x  180°/π = 57. 29577951°

                                                 = 57.30°

I.) 5π/4 is an exact radian measure. It is the same step when converting radian measures to degree measures but with denominator. 

*Here's an easy method for measuring exact radian measure. 

take first the exact radian measure without the numerator and convert it to a degree.

after that, we can just simply multiply the degree with the numerator. 

Letter L is an example of Quadrantal Angle. It is when the angles of the standard position where the terminal side lies on the x or y axis. 

I hope I covered most of our first lesson in Circular functions unit. 
might as well share my art blog here :)
[ps. hope you guys don't mind my poor grammar -_- ]