Sine Function Notes

Sine Function and Unit Circle

• The y-coordinate of a point on the unit circle corresponds to the sine of the angle.
• The x-coordinate of a point on the unit circle corresponds to the cosine of the angle.

Building Periodic Functions with the Unit Circle

• A red point's x-coordinate is the measure of the angle, and the y-coordinate is the y-coordinate of the point on the unit circle.
• As the point rotates around the unit circle, the red point traces out a curve.
• This curve is a periodic function called the sine function.
• Negative rotations (clockwise) trace out the curve in the negative direction.

Definition of Sine Function

• Denoted as $y = \sin(x)$.
• To find the sine of an angle, look at the y-coordinate of the point on the unit circle.

Calculating Sine Values

• Angle of $\frac{\pi}{6}$, y-coordinate is $\frac{1}{2}$.
• Angle of $\frac{\pi}{4}$, y-coordinate is $\frac{\sqrt{2}}{2}$.
• Angle of $\frac{\pi}{2}$, y-coordinate is 1.
• Angle of $\pi$, y-coordinate is 0.
• Angle of $\frac{3\pi}{2}$, y-coordinate is -1.
• Angle of $2\pi$, y-coordinate is 0.
• Angle of $\frac{13\pi}{6}$, y-coordinate is $\frac{1}{2}$.
• Angle of $\frac{5\pi}{2}$, y-coordinate is 1.

Graphing the Sine Function

• Plot points with x as the angle and y as the y-coordinate on the unit circle.
• Examples:
  • $\frac{\pi}{6}$ gives $\frac{1}{2}$.
  • $\frac{\pi}{4}$ gives $\frac{\sqrt{2}}{2}$.
  • $\frac{\pi}{2}$ gives 1.
  • $\pi$ gives 0.
  • $\frac{3\pi}{2}$ gives -1.
  • $2\pi$ gives 0.
  • $\frac{13\pi}{6}$ gives $\frac{1}{2}$.
  • $\frac{5\pi}{2}$ gives 1.
• The resulting curve is the sine wave.

Properties of the Sine Function

• Period: The distance it takes for the function to repeat itself, which is $2\pi$.
• Amplitude: The maximum distance the function reaches from its center line, which is 1 (ranging from -1 to 1).

Transformations of the Sine Function

• $y = \sin(2x)$: Shrinks the function in the x direction by a factor of 2, changing the period to $\pi$.
• $\frac{y}{2} = \sin(x)$: Stretches the function in the y direction by a factor of 2, changing the amplitude to 2.
• Remember to rewrite the equation as $y = 2\sin(x)$ for calculator input.
• $\frac{y}{-1} = \sin(x)$: Flips the curve over the x-axis, equivalent to $y = -1 \cdot \sin(x)$.

Summary

The sine function has a period of $2\pi$ and an amplitude of 1. Transformations can stretch or shrink the function to fit different periods and amplitudes observed in real-world phenomena. The sine wave oscillates between the y coordinates of 1 and -1. An amplifier changes the amplitude to make the sound louder.