4.4 Reference Angles

What is a Reference Angle? 

 - If an angle is in standard position, the reference angle is the acute angle formed by the terminal side of the original angle and the horizontal axis.
- In other words, the reference angle is the other half of the whole or the quadrant.
Examples:

How do you find a Reference Angle?

- You can find the value of an angle from the reference angle, and vice-versa.
- Depending on if the angle is in radians or degrees, you can find the reference angle by subtracting the horizontal axis value (in degrees or radians) from or vice-versa.
-In Quadrant I, the angle is it's own reference angle.
Examples:

  1. = 300° 

Reference Angle= 360° - 300° = 60°

You subtract  from 360° because 360° is the closest x-axis to .

 2. =


The closest x-axis to is the radian value of . Therefore, you have to subtract from because it is in Quadrant III. 

 3.
First, you have to find a co-terminal angle so it is easier to find the reference angle.



Now that you concluded is co-terminal to , you can subtract 150 from 180.


You subtract from 180 degrees because it is the closest x-axis to your co-terminal angle. 


Sources/For More Help:
-Our textbook
-http://www.mathwarehouse.com/trigonometry/reference-angle/finding-reference-angle.php
-http://www.regentsprep.org/Regents/math/algtrig/ATT3/referenceAngles.htm
-http://www.mathopenref.com/reference-angle.html
-http://www.youtube.com/watch?v=3RD-zJUj5Bo&noredirect=1

4.1 Trigonometry: Radian and Degree Measure

What is a radian?
The amount of rotation required such that the length of the intercepted arc is equal to the radius.

Angles 

• Angle-  Two rays with a common endpoint.  Angles are determined by rotating a ray about it's endpoint. 
• Initial Side- The starting position of the ray when measuring an angle
• Terminal Side- The position after the roation of a ray
• An angle can be positive or negative 
• How do we tell if it's positive or negative?  Angles are in standard position, where the vertex is at the origin, and the initial side is at the x-axis

              - Positive angles are generated by counterclockwise rotation
              - Negative angles by clockwise rotation.

• Can be measured in degrees or radians 

Angle Relationships

• Congruent- Angles with equal measures
• Complementary- Angles with measures that add up to 90

  °

   (or π/2 radians)
• Supplementary- Angles with measures that add up to 180

  °

   (or π radians)
• Coterminal- Two angles that when put in stardard position have terminal sides at the same spot

Coterminal angles 

Arc Length 

A circle has a radius of 4 inches.  Find the length of the arc intercepted by a central angle of 240°.

First, convert 240° to radians

240° = 240° ( π radians/180°)

4 π/ 3 radians

s=rθ

= 4(4π/3)

=16π/3 

= about 16.76 inches

Converting degrees to radians 

to convert degrees to radians, multiply degrees by π radians/180°( to cancel out the degrees)

135° =135 * (π radians/180°)= 3π/4 radians 

Converting radians to degrees

to convert radians to degrees, multiply degrees by 180°/π radians (to cancal out the radians) 

2 radians = 2 radians * (180°/ π radians ) = 114.59°