6.1 Graphs of the Sine and Cosine Functions

Graphs of Sine and Cosine Functions

Sine and cosine functions describe repeating behavior. They're the foundation for modeling anything that cycles: tides, sound waves, Ferris wheels, AC circuits. By adjusting a few parameters, you can control the height, width, and position of these wave-shaped graphs.

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Variations in Sine and Cosine Graphs

The general form of both functions is:

$y = a \sin(b(x - c)) + d$ or $y = a \cos(b(x - c)) + d$

Each parameter controls a specific transformation:

• $a$ controls the amplitude (vertical stretch or compression)
• $b$ controls the period (horizontal stretch or compression), calculated as $\frac{2\pi}{b}$
• $c$ controls the phase shift (horizontal translation)
• $d$ controls the vertical shift (moves the midline up or down)

Amplitude changes affect how tall or short the wave is:

• $|a| > 1$ stretches the graph vertically, making the peaks higher and the troughs lower
• $0 < |a| < 1$ compresses the graph vertically, flattening the wave
• $a < 0$ reflects the graph across the x-axis, flipping it upside down

Period changes affect how wide or narrow one full cycle is:

• $b > 1$ decreases the period, squeezing more cycles into the same horizontal space (think a high-pitched sound wave with rapid oscillations)
• $0 < b < 1$ increases the period, stretching each cycle out wider (a low-pitched sound wave with slow oscillations)

Phase shift moves the entire graph left or right:

• $c > 0$ shifts the graph to the right
• $c < 0$ shifts the graph to the left

Watch the sign carefully here. In $y = \sin(b(x - c))$, the subtraction means a positive $c$ shifts right. If you see $y = \sin(x + 3)$, rewrite it as $y = \sin(x - (-3))$, so $c = -3$ and the shift is 3 units left.

Vertical shift moves the graph up or down:

• $d > 0$ shifts the graph up
• $d < 0$ shifts the graph down

Key Features of Trigonometric Functions

Amplitude $|a|$ is the maximum vertical distance from the midline to a peak (or trough). For example, if $a = 3$, the wave reaches 3 units above and 3 units below the midline.

Period $\frac{2\pi}{b}$ is the horizontal length of one complete cycle. The frequency is the reciprocal of the period, telling you how many cycles fit in $2\pi$ units.

Midline is the horizontal line $y = d$. The graph oscillates equally above and below this line. When $d = 0$, the midline is just the x-axis.

Extrema are the maximum and minimum values the function reaches:

• Maximum value: $|a| + d$
• Minimum value: $-|a| + d$

These hold for both sine and cosine, regardless of phase shift.

Domain: All real numbers. You can plug in any x-value.

Range: $[-|a| + d, \; |a| + d]$. The output is always bounded between the minimum and maximum.

Periodic and Sinusoidal Functions

A periodic function repeats its values at regular intervals. The smallest interval over which it repeats is the period.

Sine and cosine are specifically called sinusoidal functions because of their smooth, wave-like shape. Other functions can be periodic without being sinusoidal (a sawtooth wave, for instance), but sine and cosine are the classic examples.

The input can be in degrees or radians. In pre-calculus and beyond, radians are standard because they simplify formulas and connect more naturally to the unit circle.

Transformations for Real-World Modeling

To build a sine or cosine model from a real-world situation, follow these steps:

1. Identify the periodic behavior. What quantity is repeating? (height, temperature, voltage, etc.)

2. Find the amplitude. Calculate half the distance between the maximum and minimum values: $a = \frac{\text{max} - \text{min}}{2}$

3. Find the vertical shift. The midline sits halfway between the max and min: $d = \frac{\text{max} + \text{min}}{2}$

4. Find the period and solve for $b$. Determine how long one full cycle takes, then use $b = \frac{2\pi}{\text{period}}$

5. Find the phase shift. Determine where the cycle starts relative to a standard sine or cosine curve, and set $c$ accordingly.

6. Write the function using $y = a\sin(b(x - c)) + d$ or $y = a\cos(b(x - c)) + d$

Example: Ferris Wheel

A Ferris wheel has a diameter of 40 feet, its center is 25 feet above the ground, and it completes one rotation every 60 seconds. A rider starts at the bottom.

• Amplitude: $a = 20$ (radius of the wheel)
• Vertical shift: $d = 25$ (height of the center)
• Period: 60 seconds, so $b = \frac{2\pi}{60} = \frac{\pi}{30}$
• Since the rider starts at the bottom (a minimum), you can use a negative cosine: $y = -20\cos\!\left(\frac{\pi}{30}\,x\right) + 25$

At $x = 0$, this gives $y = -20(1) + 25 = 5$ feet (the bottom of the wheel). At $x = 30$ (half a rotation), $y = -20(-1) + 25 = 45$ feet (the top). That checks out: the lowest point is $25 - 20 = 5$ ft and the highest is $25 + 20 = 45$ ft.