PRECAL Q2

• What is an angle defined as?

  An angle is a figure formed by two half lines or rays that have a common endpoint.
• What is the common endpoint of an angle called?

  The common endpoint is called the vertex.
• How is an angle defined in some other books?

  It is defined as the amount of space covered by the rotation of one of the rays.
• What is the fixed ray of an angle called?

  The fixed ray is called the initial side.
• What is the moving ray of an angle called?

  The moving ray is called the terminal side.
• What are the Greek letters used to name angles?

  • Theta = θ
  • Beta = β
  • Phi = φ
  • Omega = ω
  • Alpha = α
  • Gamma = γ
• What are the two ways to name an angle using English letters?

  Using an English capital letter with a degree symbol or an English letter with an angle sign.
• What are the two directions of angle rotation?

  The directions are clockwise and counter-clockwise.
• How is counter-clockwise rotation represented?

  Counter-clockwise rotation is represented by positive (+) direction.
• How is clockwise rotation represented?

  Clockwise rotation is represented by negative (-) direction.
• How do you draw an angle in standard position?

  1. Place the initial side on the positive x-axis.
  2. Rotate the terminal side to the desired angle.
  3. Examples:

  • 60°
  • 215°
  • -90°
  • -145°
• What defines an angle in standard position in the Cartesian plane?

  An angle is in standard position when its initial side lies on the positive x-axis and the vertex is at the origin.
• What are the units for angle measurement?

  • Degree
  • Radian
  • Revolution
• How is a degree defined?

  A degree is an angle measurement taken from dividing a whole circle into 360 parts.
• How can a degree be further divided?

  A degree can be divided into minutes and seconds.
• How many minutes are in one degree?

  1° = 60 minutes (')
• How many seconds are in one degree?

  1° = 3600 seconds ("")
• How do you convert 47.72° into DMS?

  47.72° = 47° 43' 12''
• How do you convert 55.33° into DMS?

  55.33° = 55° 19' 48''
• How do you convert 13°14'20'' into degrees?

  13°14'20'' = 13.24°
• How do you convert 28°18'36'' into degrees?

  28°18'36'' = 28.31°
• What is a radian?

  A radian is the standard unit of an angle equivalent to the length of the arc of a unit circle.
• How is a radian defined in relation to a circle?

  A radian is defined as the ratio of the length of the arc to the radius of a given circle.
• How many radians are in a full unit circle?

  2π rad = full unit circle.
• What is the equivalence of angle units?

  • 1 complete circle (or 1 revolution) = 360° = 2π rad
  • Half circle (or half revolution) = 180° = π rad
• How do you convert degrees to radians?

  Multiply degrees by $\frac{\pi \text{ rad}}{180}$.
• How do you convert radians to degrees?

  Multiply radians by $\frac{180°}{\pi \text{ rad}}$.
• What is 240° in radians?

  240° = $\frac{4\pi}{3}$ rad.
• What is 325° in radians?

  325° = $\frac{65\pi}{36}$ rad.
• What is $\frac{5\pi}{6}$ rad in degrees?


  $\frac{5\pi}{6}$ rad = 150°.
• What is $\frac{18\pi}{17}$ rad in degrees?


  $\frac{18\pi}{17}$ rad = 190.59°.
• How is the location of an angle determined in the Cartesian Plane?

  • Based on the last position of its terminal side.
  • Quadrants:
  • QI: 0° to 90°
  • QII: 90° to 180°
  • QIII: 180° to 270°
  • QIV: 270° to 360°
• What is a reference angle?

  • The smallest positive acute angle measured from the terminal side of a given angle to the nearest x-axis.
  • Cases for degrees:
  • QI: RA = θ
  • QII: RA = 180° - θ
  • QIII: RA = θ - 180°
  • QIV: RA = 360° - θ
• What is the reference angle in radians?

  • Cases for radians:
  • QI: RA = θ
  • QII: RA = π - θ
  • QIII: RA = θ - π
  • QIV: RA = 2π - θ
• What is the reference angle if the given angle is 173°?

  The reference angle is 7° (RA = 180° - 173°).
• What is the reference angle if the given angle is 245°?

  The reference angle is 65° (RA = 245° - 180°).
• What is the reference angle if the given angle is 305°?

  The reference angle is 55° (RA = 360° - 305°).
• How do you find the reference angle for an angle greater than 360°?

  Deduct complete revolutions until the angle is less than 360° and then apply Cases 1-4.
• What are coterminal angles?

  • Angles that have the same terminal and initial sides.
  • To find coterminal angles:
  • Counter-clockwise: \( \theta + 360°(n) \) or \( \theta + 2\pi(n) \)
  • Clockwise: \( \theta - 360°(n) \) or \( \theta - 2\pi(n) \)
• What is a coterminal angle for 45° in the counter-clockwise direction?

  105° (45° + 360°(1)).