Sketching Angles and Reference Angles

Jul 2, 2024

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Lecture on Sketching Angles and Reference Angles

Introduction

  • Discussion on sketching angles and their reference angles.
  • Angle measurement and rotation.

Angle Measurement

  • Counterclockwise Rotation: Angle has a positive measure.
  • Clockwise Rotation: Angle has a negative measure.
  • Example: Positive Angle
    • Graphed at around +40 degrees.
    • Full rotation = 360 degrees.
  • Example: Negative Angle
    • Graphed at around -200 degrees.
  • Components of an Angle:
    • Initial Side: Starting side of the angle.
    • Terminal Side: Ending side of the angle.
    • Vertex: Point where two sides meet.

Standard Position

  • Defined as positioning the initial side along the positive x-axis in an x, y coordinate system.
  • Example: Sketching 60 Degrees in Standard Position
    • Rotate in the counterclockwise direction.
    • Initial side along the x-axis, vertex at the origin.
    • Terminal side at about +60 degrees.
  • Example: Sketching -150 Degrees in Standard Position
    • Rotate in the clockwise direction.
    • Initial side along the x-axis, vertex at the origin.
    • Terminal side at about -150 degrees.

Reference Angles

  • Definition: An acute angle formed by the terminal side and the x-axis.
    • Acute angle: Between 0 and 90 degrees (or 0 and Ο€/2 radians).
  • Example: Sketching 120 Degrees and Identifying Reference Angle
    • Positive direction rotation.
    • Terminal side about +120 degrees.
    • Reference angle with the x-axis: 60 degrees (180 - 120 = 60).
  • Importance: Essential in trigonometry for graphing angles accurately.

Sketching in Radians

  • Example: 5Ο€/6 Radians
    • Convert radians to degrees: Ο€/6 β‰ˆ 30 degrees.
    • 5Ο€/6 β‰ˆ 150 degrees (180 - 30).
    • Reference angle: Ο€/6 radians.
  • Conversion Memorization: Key Angles
    • Ο€ = 180 degrees, 2Ο€ = 360 degrees.
    • Ο€/6 = 30 degrees, Ο€/4 = 45 degrees, Ο€/3 = 60 degrees.
    • Essential to memorize these conversions.

More on Reference Angles

  • Example: -5Ο€/3 Radians
    • Negative rotation (clockwise).
    • Close to -2Ο€ (negative full rotation).
    • Sketch: -5Ο€/3 = -300 degrees (360 - 60).
    • Reference angle: Ο€/3 radians (60 degrees).
  • Key Points of Reference Angles:
    • Always a positive acute angle.
    • The positive angle between terminal side and x-axis.

Conclusion

  • Importance of understanding standard position and reference angles.
  • Crucial for upcoming topics in trigonometry.