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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.
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