math Ch5.1

Chapter Overview

• Chapter 5: Trigonometric Functions: Unit Circle Approach

  • Provides a comprehensive understanding of the unit circle and its application to trigonometric functions.

5.1 The Unit Circle

• Objectives:

  • Understanding the unit circle

  • Identifying terminal points on the unit circle

  • Utilizing reference numbers

The Unit Circle

• Definition: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate system.

  • Mathematically, the equation is:

    • ( x^2 + y^2 = 1 )

• Key Properties:

  • Points on the unit circle represent coordinates where the distance from the origin is always 1.

  • Example: Consider a point that satisfies the equation confirming it lies on the circle.

Terminal Points on the Unit Circle

• Definition: Terminal points are points obtained by moving a distance t along the unit circle starting from (1, 0).

  • For positive t: Counterclockwise movement

  • For negative t: Clockwise movement

• The resultant point P(x, y) is the terminal point determined by the angle t.

  • Circumference Calculation:

    • Circumference, C = 2π(1) = 2π

    • Moving halfway (π) or a quarter (π/2) around the circle yields well-defined coordinates.

Finding Terminal Points

• Example: Finding terminal points for specific t values:

  • For ( t = 3π ) and ( t = −π ), the coordinates provide insight into the behavior of trigonometric functions around the unit circle.

• Symmetry: The unit circle exhibits symmetry across x and y axes, aiding in determining terminal points in quadrants II, III, and IV using values from quadrant I.

Reference Number

• A tool to simplify finding terminal points regarding their position relative to the x-axis.

  • Definition: The reference number is the acute angle that lies between the terminal point and the x-axis.

• Finding Reference Numbers:

  • Understanding in which quadrant the terminal point lies is crucial to determining the reference number.

Example Work

• Example 5: Calculation of reference numbers based on given angles to determine terminal points effectively using established frameworks.

• Example 6: Demonstrates utilizing reference numbers to navigate through quadrant characteristics and signify the appropriate signs (+/-) for x and y coordinates.

Circular Behavior of Angles

• The unit circle maintains periodic properties where angles can be increased or decreased by multiples of 2π without changing the terminal point location.

• Example 7: Illustrates finding terminal points for larger angles by reducing them to an equivalent angle within the standard unit circle range.

Conclusion

• Understanding the unit circle, terminal points, and reference numbers equips students with the necessary tools to handle trigonometric functions and their applications.