Lecture Notes: The Unit Circle and Trigonometric Functions

The Unit Circle

• Definition: A unit circle is a circle with a radius of 1.
  • The hypotenuse of a triangle inscribed in the unit circle will also have a value of 1 (radius = hypotenuse).
  • The distance between the center of the circle and a point on the circle is 1.

Triangle in the Unit Circle

• Components:

  • x-axis: Represents the x-coordinate.
  • y-axis: Represents the y-coordinate.
  • Angle (θ): Located inside the triangle, measured in degrees.

• Example: For θ = 30 degrees:

  • Coordinates:
    • x-coordinate = ( \frac{\sqrt{3}}{2} )
    • y-coordinate = ( \frac{1}{2} )

Trigonometric Functions on the Unit Circle

• Sine (sin θ):

  • Defined as the y-coordinate on the unit circle.
  • For a unit circle, where radius (r) = 1, it simplifies to:
    • sin 30 = ( \frac{1}{2} )

• Cosine (cos θ):

  • Defined as the x-coordinate on the unit circle.
  • For the example:
    • cos 30 = ( \frac{\sqrt{3}}{2} )

Additional Trigonometric Functions

• Tangent (tan θ):

  • Formula: ( \text{tan θ} = \frac{\text{sin θ}}{\text{cos θ}} = \frac{y}{x} )

• Cosecant (csc θ):

  • Formula: ( \text{csc θ} = \frac{1}{y} )

• Secant (sec θ):

  • Formula: ( \text{sec θ} = \frac{1}{x} )

• Cotangent (cot θ):

  • Formula: ( \text{cot θ} = \frac{x}{y} )

Important Notes

• The above definitions and relationships hold true specifically for a unit circle (r = 1).
• If the radius is not 1, equations will be adjusted accordingly, but this topic will be discussed later.

Conclusion

• Familiarize with all six trigonometric functions in the context of the unit circle.
• Remember the basic relationships and how they apply when the radius equals one.