Graphing Sine and Cosine Functions
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn how to graph the sine and cosine functions, identify their amplitude, period, and midline, and understand their basic shape.
The Concept
Section titled “The Concept”The sine and cosine functions are periodic. They repeat the same pattern over and over, like a wave.
Standard sine function y = sin(x):
- Amplitude = 1 (height from midline to peak)
- Period = 2π (one full cycle)
- Midline = y = 0
- Starts at (0, 0), peaks at (π/2, 1), crosses zero at (π, 0), bottoms out at (3π/2, -1), returns to (2π, 0)
Standard cosine function y = cos(x):
- Amplitude = 1
- Period = 2π
- Midline = y = 0
- Starts at (0, 1), crosses zero at (π/2, 0), bottoms out at (π, -1), crosses zero at (3π/2, 0), returns to (2π, 1)
Both functions oscillate between -1 and 1. They’re actually the same shape, just shifted: cos(x) = sin(x + π/2). Cosine is sine slid π/2 to the left.
Worked Example
Section titled “Worked Example”1. Key points for y = sin(x) over one period (0 to 2π)
| x | 0 | π/2 | π | 3π/2 | 2π |
|---|---|---|---|---|---|
| sin(x) | 0 | 1 | 0 | -1 | 0 |
Plot these five points and connect them with a smooth curve. The shape is a wave that starts at zero, goes up, comes back, goes down, and returns.
2. Key points for y = cos(x) over one period
| x | 0 | π/2 | π | 3π/2 | 2π |
|---|---|---|---|---|---|
| cos(x) | 1 | 0 | -1 | 0 | 1 |
Same wave shape, but it starts at the top instead of the middle.
3. Comparing sin and cos at specific values
- At x = 0: sin(0) = 0, cos(0) = 1
- At x = π/2: sin(π/2) = 1, cos(π/2) = 0
- At x = π: sin(π) = 0, cos(π) = -1
- At x = 3π/2: sin(3π/2) = -1, cos(3π/2) = 0
They trade off. When one is at its peak, the other is at zero.
Real-World Application
Section titled “Real-World Application”Sine and cosine graphs model many real phenomena:
- Sound waves and music (frequency determines pitch, amplitude determines volume)
- Alternating current (AC) electricity
- Seasonal temperature changes over a year
- Tides and ocean waves
- Spring motion and vibrations
- Daylight hours throughout the year
Example: the height of a tide can be modeled with a sine or cosine function. The amplitude represents how high the tide rises, and the period represents the time between high tides (roughly 12.4 hours).