Calculating Arc Length in Degrees

• angle / total angle of circle = arc length of angle / total circumference of circle
  • θ / 360° = arc length / 2pi*r
  • by multiplying both sides by 2pi*r, we solve for arc length
  • arc length = (2pi*r)*(θ / 360°)

Introduction to Radians

• radians measure the arc length of the angle created
• measuring radians uses a circle with a radius of 1
  • so the arc length of the entire circle (aka the circumference) is 2pi*r, but r = 1, so it's just 2pi
• 2pi radians = 360°
  • from this equality you can figure out many other angle measures
  • pi radians = 180°
  • 1 radian = 180°/pi
  • 1° = pi/180 radians
• the unit of an angle in radians often isn't shown, for example if θ = 3, that would represent 3 radians

Converting Angles from Degrees to Radians

• because 2pi radians = 360°, divide both sides by 360 to become 1° = pi/180 radians
• if 1° = pi/180 radians, then x degrees is x*(pi/180)
  • 160° = 160(pi/180) radians = 8pi/9 radians
  • 195° = 195(pi/180) radians = 13pi/12 radians
  • 280° = 280(pi/180) radians = 14pi/9 radians

Converting Angles from Radians to Degrees

• because 2pi radians = 360°, divide both sides by 2pi to become 1 radian = 180°/pi
• if 1 radian = 180°/pi, then x radians is x*(180°/pi)
  • 1/4 radians = (1/4)*(180°/pi) = 14°
  • 3pi/2 radians = (3pi/2)*(180°/pi) = 270°
  • 5pi/18 radians = (5pi/18)*(180°/pi) = 50°

Calculating Arc Length in Radians

• with degrees, we had arc length = (2pi*r)*(θ / 360°)
• with radians, θ will be measured in radians and replace 360° with 2pi
• 2pi cancels out, so arc length = θ*r

Coterminal Angles

• angles with the same terminal arm (their terminal arms overlap)
• add or subtract a full rotation (360° or 2pi) to any angle to find another coterminal angle
  • θ = n*360° or θ = n*2pi, where n is an integer (-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5)
• angles that are coterminal have the EXACT same trigonometric ratios

The Unit Circle

• the angles 0°, 30°, 45°, 60°, and 90° (or 0, pi/6, pi/4, pi/3, and pi/2), all have exact trigonometric ratios

• angles in quadrant 2, 3, and 4 have the SAME trigo ratios as those in quadrant 1 (except they may be negative)
  • in quadrant 1, all trig ratios are positive
  • in quadrant 2, only sin is positive (cos and tan are negative)
  • in quadrant 3, only tan is positive (sin and cos are negative)
  • in quadrant 4, only cos is positive (sin and tan and negative)
  • this is summarized in ASTC (All Students Take Calculus

Degree Mode and Radian Mode