Plotting Key Points of $y = \cos(x)$

Graph one complete cycle of $y = \cos(x)$ from $x = 0$ to $x = 2\pi$.

Find the starting point: $\cos(0) = 1$ = Point: $(0, 1)$

Find the first zero: $\cos\left(\frac{\pi}{2}\right) = 0$ = Point: $\left(\frac{\pi}{2}, 0\right)$

Find the minimum: $\cos(\pi) = -1$ = Point: $(\pi, -1)$

Find the second zero: $\cos\left(\frac{3\pi}{2}\right) = 0$ = Point: $\left(\frac{3\pi}{2}, 0\right)$

Find the endpoint (back to max): $\cos(2\pi) = 1$ = Point: $(2\pi, 1)$

Answer: The five key points are $(0, 1)$, $\left(\frac{\pi}{2}, 0\right)$, $(\pi, -1)$, $\left(\frac{3\pi}{2}, 0\right)$, and $(2\pi, 1)$. Connect these with a smooth wave.

Graphing $y = 3\cos(x)$

How does the graph of $y = 3\cos(x)$ differ from $y = \cos(x)$?

Identify the amplitude: The coefficient $3$ multiplies all $y$-values = Amplitude $= 3$

Find the new maximum: $3 \times 1 = 3$ = Maximum at $y = 3$

Find the new minimum: $3 \times (-1) = -3$ = Minimum at $y = -3$

Check the period: No coefficient on $x$, so period unchanged = Period $= 2\pi$

Answer: The graph of $y = 3\cos(x)$ is a vertical stretch of $y = \cos(x)$. It oscillates between $-3$ and $3$ instead of $-1$ and $1$, but the period remains $2\pi$.

Graphing $y = \cos(2x)$

Find the period of $y = \cos(2x)$ and describe how its graph differs from $y = \cos(x)$.

Identify the coefficient of $x$: The coefficient is $B = 2$ = $B = 2$

Calculate the period: $\text{Period} = \frac{2\pi}{B} = \frac{2\pi}{2}$ = Period $= \pi$

Interpret the result: One complete cycle now takes $\pi$ instead of $2\pi$ = Graph is compressed horizontally

Check the amplitude: No coefficient in front, so amplitude $= 1$ = Amplitude unchanged

Answer: The graph of $y = \cos(2x)$ has period $\pi$, meaning it completes two full cycles in the same space where $y = \cos(x)$ completes one. The amplitude stays at $1$.

Graphing $y = \cos(x - \frac{\pi}{3})$

Describe the transformation in $y = \cos\left(x - \frac{\pi}{3}\right)$.

Identify the phase shift form: The form is $y = \cos(x - C)$ where $C = \frac{\pi}{3}$ = Phase shift $= \frac{\pi}{3}$

Determine the direction: Minus $C$ means shift to the RIGHT = Shift right by $\frac{\pi}{3}$

Find the new starting point: Maximum was at $x = 0$, now at $x = \frac{\pi}{3}$ = New max: $\left(\frac{\pi}{3}, 1\right)$

Verify other key points shift: Zero was at $\frac{\pi}{2}$, now at $\frac{\pi}{2} + \frac{\pi}{3} = \frac{5\pi}{6}$ = All points shift right

Answer: The graph of $y = \cos\left(x - \frac{\pi}{3}\right)$ is the same as $y = \cos(x)$ shifted $\frac{\pi}{3}$ units to the right. The maximum occurs at $x = \frac{\pi}{3}$ instead of $x = 0$.

Mistake: Confusing sine and cosine starting points

Why: Sine starts at $0$ (going up), while cosine starts at its maximum $1$. Both are sinusoids but with different starting positions.

Correct: Remember: $\cos(0) = 1$ (starts at max), $\sin(0) = 0$ (starts at zero)

Mistake: Getting phase shift direction wrong

Why: The equation $y = \cos(x - C)$ shifts RIGHT by $C$, not left. The minus sign is counterintuitive.

Correct: $y = \cos(x - C)$ shifts RIGHT; $y = \cos(x + C)$ shifts LEFT

Mistake: Forgetting to divide $2\pi$ by $B$ for period

Why: Students sometimes think $y = \cos(2x)$ has period $2$, but period $= \frac{2\pi}{B}$.

Correct: For $y = \cos(Bx)$, period $= \frac{2\pi}{B}$, not $B$

Modeling Daily Temperature

Daily temperature follows a cosine pattern, with maximum temperature in the afternoon and minimum at night.

If the average temperature is 20 degrees Celsius with a 10 degree variation, and maximum occurs at 3 PM, the model is: $T(t) = 20 + 10\cos\left(\frac{\pi}{12}(t - 15)\right)$ where $t$ is hours after midnight.

Sound and Music

Musical notes are produced by sound waves that follow sinusoidal patterns. The cosine function models the pressure variation.

The note A4 (440 Hz) can be modeled as $P(t) = A\cos(880\pi t)$ where $t$ is time in seconds and $A$ is amplitude.

Q: How is cosine different from sine?

A: Cosine and sine have the same shape but different starting points. Cosine starts at its maximum ($\cos(0) = 1$), while sine starts at zero ($\sin(0) = 0$). Mathematically, $\cos(x) = \sin\left(x + \frac{\pi}{2}\right)$.

Q: Why is the period $2\pi$?

A: The period is $2\pi$ because cosine is defined using the unit circle, and going around the circle once (360 degrees or $2\pi$ radians) brings you back to the starting point.

Q: What does a negative amplitude mean?

A: A negative amplitude like $y = -2\cos(x)$ reflects the graph across the $x$-axis. The wave is flipped upside down: it starts at $-2$ (minimum) instead of $2$ (maximum).