Exploring Trigonometry Functions and Graphs

Aug 30, 2024

Trigonometry Functions Lecture Notes

Introduction

  • Discussed trigonometry functions, starting with basic angles.
  • How to use reference angles to find trigonometry functions of any angle.
  • Graphing trigonometry functions and understanding their behavior.

Angles

  • An angle is typically measured in reference to the x-axis.
  • Initial side: where the angle starts.
  • Terminal side: where the angle ends.
    • Positive angles: counterclockwise rotation.
    • Negative angles: clockwise rotation.

Measuring Angles

  • Two main systems: Degrees and Radians.
    • Radians involve pi, an irrational number representing the relationship between a circle's diameter and circumference.
    • Degrees are more commonly used for everyday angle measurement.

Conversion between Radians and Degrees

  • Equation: 2π radians = 360 degrees.
    • Convert degrees to radians: multiply by π/180.
    • Convert radians to degrees: multiply by 180/π.

Graphing Angles

  • Understanding angles in a unit circle and graphing on the XY-axis.
  • Co-terminal angles: Angles with the same terminal side.

Trigonometry Functions

  • Trig functions relate the angles of a right triangle to ratios of its sides.
    • Sine (sin): Opposite over Hypotenuse.
    • Cosine (cos): Adjacent over Hypotenuse.
    • Tangent (tan): Opposite over Adjacent.
    • Cosecant (csc): Hypotenuse over Opposite (reciprocal of sin).
    • Secant (sec): Hypotenuse over Adjacent (reciprocal of cos).
    • Cotangent (cot): Adjacent over Opposite (reciprocal of tan).

Unit Circle and Trigonometry

  • Unit Circle: Radius of one.
    • The x-coordinate represents cos, and the y-coordinate represents sin.
    • Tangent: y/x, or sin/cos.

Quadrants and Sign of Trigonometry Functions

  • Mnemonic: All Students Take Calculus (ASTC)
    • Quadrant I: All trig functions positive.
    • Quadrant II: Only sin and cosec are positive.
    • Quadrant III: Only tan and cot are positive.
    • Quadrant IV: Only cos and sec are positive.

Reference Angles

  • A reference angle is the acute angle that the terminal side of a given angle makes with the x-axis.
  • Used to find trig functions of any angle by assessing the quadrant location and using ASTC.

Trigonometry Identities

  • Important for solving equations and expressions.
    • Sine and Cosine relationship: sin²θ + cos²θ = 1.
    • Tangent identity: tanθ = sinθ/cosθ.

Graphing Trigonometry Functions

  • Understand the structure of sine, cosine, and tangent graphs.
  • Amplitude and Period: Key Concepts
    • Amplitude: Height from the center line to the peak.
    • Period: How long it takes for the function to repeat.

Transformations

  • Vertical Stretch or Compression: Changes to amplitude.
  • Horizontal Stretch or Compression: Changes to period.
  • Phase Shift (Horizontal Shift): Based on modifications inside the function argument.
    • Positive indicates left shift, negative indicates right shift.

Practice with Examples

  • Applying concepts of amplitude, period, and shifts to graph sine and cosine functions.
  • Ensures understanding of translation and how it affects graph placement.

Conclusion

  • Reinforce the importance of understanding unit circle values.
  • Encouragement to review trigonometry identities for calculus applications.