Cosine Function: Comprehensive Notes

Cosine Function

• Definition: The cosine of an angle is the x-coordinate of the point where the terminal side of the angle intersects the unit circle.
  • Denoted as $cos(\theta)$, where $\theta$ is the angle.
  • If a point on the unit circle has coordinates $(u, v)$, then $cos(\theta) = u$ and $sin(\theta) = v$.

Unit Circle and Cosine

• To find the cosine of an angle (e.g., $\frac{\pi}{2}$):
  • Visualize the unit circle.
  • Locate the angle on the unit circle.
  • Identify the x-coordinate of that point. For $\frac{\pi}{2}$, the point is $(0, 1)$, so $cos(\frac{\pi}{2}) = 0$.

Example: Cosine of $\frac{5\pi}{6}$

• Locate $\frac{5\pi}{6}$ on the unit circle.

• Identify the coordinates of the point: $(-\frac{\sqrt{3}}{2}, \frac{1}{2})$.

• The cosine is the x-coordinate: $cos(\frac{5\pi}{6}) = -\frac{\sqrt{3}}{2}$.

Graphing the Cosine Function

• To graph $y = cos(x)$, plot the cosine values for various angles $x$.

• Key points:

  • $x = 0$, $cos(0) = 1$
  • $x = \frac{\pi}{2}$, $cos(\frac{\pi}{2}) = 0$
  • $x = \pi$, $cos(\pi) = -1$
  • $x = \frac{3\pi}{2}$, $cos(\frac{3\pi}{2}) = 0$
  • $x = 2\pi$, $cos(2\pi) = 1$
  • $x = \frac{5\pi}{2}$, $cos(\frac{5\pi}{2}) = 0$
  • $x = 3\pi$, $cos(3\pi) = -1$

Characteristics of the Cosine Graph

• Periodic Function: Repeats its values at regular intervals.

• Period: The horizontal distance for one complete cycle. For $y = cos(x)$, the period is $2\pi$.

• Amplitude: The distance from the midline to the maximum (or minimum) value. For $y = cos(x)$, the amplitude is 1.

• Key Difference from Sine: The cosine function starts at its maximum value (1) at $x = 0$, while the sine function starts at the origin.

Transformations of the Cosine Function

• Horizontal Stretch/Compression:

  • If the function is $y = cos(\frac{x}{5})$, the graph is stretched horizontally by a factor of 5. If a point on the standard cosine function is $(2\pi, 1)$, then the corresponding point on the transformed function is $(10\pi, 1)$.
  • The period becomes $10\pi$.

• Vertical Stretch/Compression:

  • If the function is $y = 6cos(x)$, the graph is stretched vertically by a factor of 6.
  • The amplitude becomes 6.

General Form and Calculator Use

• The general form is $y = Acos(Bx)$, where:

  • A affects the amplitude (vertical stretch).
  • B affects the period (horizontal stretch/compression).

• Using a graphing calculator:

  • Changing the value of A stretches or shrinks the graph in the y-direction.
  • Increasing B compresses the period (makes it smaller).
  • Decreasing B stretches the period (makes it bigger).

Sine vs. Cosine

• Both are basic periodic functions.

• Sine starts at the origin, while cosine starts at its maximum point.