Trigonometric functions notes

Trigonometric Functions and the Unit Circle

Introduction to Sine and Cosine

• Unit Circle Representation:
  • Points on the unit circle are represented as coordinates (x, y).
  • The angle is measured counter-clockwise from the positive x-axis, which is considered positive.
  • Clockwise direction is considered negative.

Key Points on the Unit Circle

• (1, 0) corresponds to 0 degrees.
• (0, 1) corresponds to 90 degrees.
• (-1, 0) corresponds to 180 degrees.
• (0, -1) corresponds to 270 degrees.

Cosine and Sine Definitions

• \cos(\theta) = x, where x is the x-coordinate of the point on the unit circle.
• \sin(\theta) = y, where y is the y-coordinate of the point on the unit circle.

Special Angles

• 45-45-90 Triangle:

  • For a 45-45-90 triangle, if the legs are of length 1, the hypotenuse is \sqrt{2}.
  • \sin(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} and \cos(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}.

• 30-60-90 Triangle:

  • For a 30-60-90 triangle:
    • \sin(30^\circ) = \frac{1}{2}
    • \cos(30^\circ) = \frac{\sqrt{3}}{2}
    • \sin(60^\circ) = \frac{\sqrt{3}}{2}
    • \cos(60^\circ) = \frac{1}{2}

General Representation

• A point on the unit circle can be represented as (\cos A, \sin A).

Trigonometric Ratios

• \sin = \frac{\text{opposite}}{\text{hypotenuse}}
• \cos = \frac{\text{adjacent}}{\text{hypotenuse}}

Amplitude and Period

• Period: The horizontal distance required for the function to complete one full cycle.
• Amplitude: The vertical distance from the midline to the maximum or minimum value of the function. It represents the height of the function.