Plotting the Basic Sine Curve

Graph $y = \sin(x)$ for $0 \leq x \leq 2\pi$

Create a table of key values: Identify the five key points: $x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$ = Key x-values identified

Calculate the y-values: $\sin(0)=0$, $\sin(\frac{\pi}{2})=1$, $\sin(\pi)=0$, $\sin(\frac{3\pi}{2})=-1$, $\sin(2\pi)=0$ = Points: $(0,0)$, $(\frac{\pi}{2},1)$, $(\pi,0)$, $(\frac{3\pi}{2},-1)$, $(2\pi,0)$

Plot the points: Mark each point on the coordinate plane = Five points plotted

Connect with a smooth curve: Draw a smooth wave connecting all points = One complete sine wave

Answer: The graph starts at the origin, rises to a maximum of 1, returns to 0, falls to a minimum of -1, and returns to 0 to complete one cycle.

Graphing with Amplitude Change

Graph $y = 3\sin(x)$ and describe how it differs from $y = \sin(x)$

Identify the amplitude: The coefficient 3 in front of sin is the amplitude = Amplitude = 3

Calculate the new key points: Multiply all y-values by 3: $(0,0)$, $(\frac{\pi}{2},3)$, $(\pi,0)$, $(\frac{3\pi}{2},-3)$, $(2\pi,0)$ = Key points with amplitude 3

Note the range: The function oscillates between $-3$ and $3$ = Range: $[-3, 3]$

Draw the graph: The wave is stretched vertically, 3 times taller = Graph complete

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

Graphing with Period Change

Graph $y = \sin(2x)$ and find its period

Find the period formula: Period $= \frac{2\pi}{|B|}$ where $B$ is the coefficient of $x$ = Period formula identified

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

Find key points for one period: Divide $[0, \pi]$ into four equal parts: $0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi$ = Key x-values identified

Calculate y-values: $\sin(0)=0$, $\sin(\frac{\pi}{2})=1$, $\sin(\pi)=0$, $\sin(\frac{3\pi}{2})=-1$, $\sin(2\pi)=0$ = Points: $(0,0)$, $(\frac{\pi}{4},1)$, $(\frac{\pi}{2},0)$, $(\frac{3\pi}{4},-1)$, $(\pi,0)$

Answer: The graph of $y = \sin(2x)$ completes one full cycle in $\pi$ radians instead of $2\pi$. It is horizontally compressed, completing two cycles in the same interval where $\sin(x)$ completes one.

General Sinusoidal Function

Describe the graph of $y = 2\sin(x - \frac{\pi}{4}) + 1$

Identify the amplitude: The coefficient of sine is 2 = Amplitude = 2

Find the period: The coefficient of $x$ is 1, so Period $= \frac{2\pi}{1} = 2\pi$ = Period = $2\pi$

Find the phase shift: The expression $(x - \frac{\pi}{4})$ shifts right by $\frac{\pi}{4}$ = Phase shift = $\frac{\pi}{4}$ right

Find the vertical shift: The $+1$ at the end shifts the midline up by 1 = Vertical shift = 1 up

Determine the range: Midline at $y=1$, amplitude 2, so range is $[1-2, 1+2] = [-1, 3]$ = Range: $[-1, 3]$

Answer: The graph has amplitude 2, period $2\pi$, is shifted right by $\frac{\pi}{4}$, and shifted up by 1. It oscillates between $-1$ and $3$ with its midline at $y = 1$.

Mistake: Confusing degrees and radians on the x-axis

Why: The standard sine graph uses radians. $\pi \approx 3.14$ radians equals 180 degrees.

Correct: Always check if your calculator and graph are in the same mode. For calculus and higher math, radians are standard.

Mistake: Getting the phase shift direction wrong

Why: In $y = \sin(x - c)$, the shift is to the RIGHT, not left. The minus inside creates opposite behavior.

Correct: $y = \sin(x - \frac{\pi}{2})$ shifts RIGHT by $\frac{\pi}{2}$. $y = \sin(x + \frac{\pi}{2})$ shifts LEFT by $\frac{\pi}{2}$.

Mistake: Confusing amplitude with range

Why: Amplitude is the distance from the midline to the peak, not the total height.

Correct: For $y = 3\sin(x)$, amplitude is 3, but the total height (range) is 6 (from $-3$ to $3$).

Mistake: Forgetting that the coefficient of $x$ affects period, not amplitude

Why: In $y = \sin(Bx)$, $B$ compresses or stretches horizontally, changing the period.

Correct: Period $= \frac{2\pi}{|B|}$. Larger $B$ means shorter period (faster oscillation).

Sound Waves and Music

Pure musical tones are sine waves. A tuning fork vibrates at 440 Hz (A note), creating a sine wave in air pressure.

The equation $y = \sin(880\pi t)$ models a 440 Hz sound wave, where $t$ is time in seconds.

Electrical Engineering - AC Current

Household electricity uses alternating current that follows a sine wave pattern.

European outlets provide 230V at 50 Hz: $V(t) = 325\sin(100\pi t)$ where the peak voltage is about 325V.

Ocean Tides

Tidal heights follow a roughly sinusoidal pattern due to the moon's gravitational pull.

If high tide is at 6 AM with height 3 meters and low tide is at 12 PM with height 1 meter, the tide can be modeled with a sine function.

Q: Why does the sine function start at zero?

A: On the unit circle, sine represents the y-coordinate. At angle 0 (pointing right on the x-axis), the y-coordinate is 0. This is why $\sin(0) = 0$.

Q: What is the difference between sine and cosine graphs?

A: The cosine graph is the same shape as sine, but shifted left by $\frac{\pi}{2}$. In fact, $\cos(x) = \sin(x + \frac{\pi}{2})$. Cosine starts at its maximum (1) while sine starts at 0.

Q: How do I graph negative amplitude like $y = -2\sin(x)$?

A: A negative amplitude reflects the graph across the x-axis. The wave that normally goes up first now goes down first. The peaks and troughs are inverted.