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

Cosine and Sine Definitions

  • cos⁡(θ)=x\cos(\theta) = x, where xx is the x-coordinate of the point on the unit circle.
  • sin⁡(θ)=y\sin(\theta) = y, where yy 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 2\sqrt{2}.
    • sin⁡(45∘)=12=22\sin(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} and cos⁡(45∘)=12=22\cos(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}.

  • 30-60-90 Triangle:

    • For a 30-60-90 triangle:
      • sin⁡(30∘)=12\sin(30^\circ) = \frac{1}{2}
      • cos⁡(30∘)=32\cos(30^\circ) = \frac{\sqrt{3}}{2}
      • sin⁡(60∘)=32\sin(60^\circ) = \frac{\sqrt{3}}{2}
      • cos⁡(60∘)=12\cos(60^\circ) = \frac{1}{2}

General Representation

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

Trigonometric Ratios

  • sin⁡=oppositehypotenuse\sin = \frac{\text{opposite}}{\text{hypotenuse}}
  • cos⁡=adjacenthypotenuse\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.