Unit Circle and Trigonometric Functions

Understanding the Unit Circle

• The unit circle is a circle centered at the origin with a radius of 1.
• Important coordinates on the unit circle:
  • Intersection with x-axis: ( (1, 0) )
  • Intersection above the origin: ( (0, 1) )
  • Intersection on the left: ( (-1, 0) )
  • Intersection below the origin: ( (0, -1) )

Angles and Conventions

• Positive Angles: Drawn counterclockwise from the initial side along the positive x-axis to the terminal side.
• Negative Angles: Drawn clockwise.

Creating a Right Triangle

• To explore trigonometric functions, drop an altitude from the terminal side of the angle to the x-axis to form a right triangle.
• Hypotenuse: Radius of the unit circle, length = 1.
• Base: The x-coordinate of the intersection point (denoted as (a)).
• Height: The y-coordinate of the intersection point (denoted as (b)).

Extending Trigonometric Definitions

• Using SOHCAHTOA:
  • Cosine (cos) is adjacent over hypotenuse: ( \cos(\theta) = \frac{a}{1} = a )
  • Sine (sin) is opposite over hypotenuse: ( \sin(\theta) = \frac{b}{1} = b )

New Definition of Trigonometric Functions

• Cosine: The x-coordinate where the terminal side of the angle intersects the unit circle.
• Sine: The y-coordinate where the terminal side of the angle intersects the unit circle.
• Tangent (tan): Defined as the sine of theta over the cosine of theta:
  • ( \tan(\theta) = \frac{b}{a} )

Advantages of the Unit Circle Definition

• Extends beyond the limitations of traditional SOHCAHTOA.
• Applicable for angles less than 0 and greater than 90 degrees.
• Allows for consistent definitions without needing a right triangle.

Summary and Next Steps

• The unit circle provides a way to define trigonometric functions for any angle.
• Further videos will explore using these definitions to evaluate trigonometric ratios.