Graphs of Sine and Cosine

The graphs of sine and cosine are wave patterns that repeat forever. Understanding these waves is essential for AC circuit analysis in electrical work, sound and vibration in physics, and any application involving periodic (repeating) behavior.

The Basic Sine Graph

The function $y = \sin(x)$ produces a smooth wave that oscillates between $-1$ and $1$. One complete cycle takes $2\pi$ radians (360°).

Key features of $y = \sin(x)$:

• Starts at 0 when $x = 0$
• Reaches maximum of 1 at $x = \frac{\pi}{2}$
• Returns to 0 at $x = \pi$
• Reaches minimum of −1 at $x = \frac{3\pi}{2}$
• Completes the cycle at $x = 2\pi$

The Basic Cosine Graph

The function $y = \cos(x)$ produces the same wave shape, but shifted to the left by $\frac{\pi}{2}$ — it starts at its maximum value.

Key features of $y = \cos(x)$:

• Starts at 1 when $x = 0$
• Crosses zero at $x = \frac{\pi}{2}$
• Reaches minimum of −1 at $x = \pi$
• Crosses zero at $x = \frac{3\pi}{2}$
• Returns to 1 at $x = 2\pi$

The sine and cosine graphs are identical in shape — cosine is just sine shifted left by $\frac{\pi}{2}$:

$\cos(x) = \sin\!\left(x + \frac{\pi}{2}\right)$

The General Form: Transformations

The general form for both sine and cosine functions is:

$y = A \sin(B(x - C)) + D \qquad \text{or} \qquad y = A \cos(B(x - C)) + D$

Each parameter controls a specific transformation:

Amplitude ($A$)

The amplitude is $|A|$ — the distance from the center line to the peak (or trough). It determines how “tall” the wave is.

• $y = 3\sin(x)$ has amplitude 3 — the wave goes from $-3$ to $3$
• $y = 0.5\sin(x)$ has amplitude 0.5 — the wave goes from $-0.5$ to $0.5$
• If $A$ is negative, the wave is flipped upside down

Period ($B$)

The period is the horizontal distance for one complete cycle:

$\text{Period} = \frac{2\pi}{|B|}$

• $y = \sin(2x)$ has period $\frac{2\pi}{2} = \pi$ — the wave completes twice as fast
• $y = \sin\!\left(\frac{x}{2}\right)$ has period $\frac{2\pi}{1/2} = 4\pi$ — the wave stretches out

Phase Shift ($C$)

The phase shift moves the graph left or right. Positive $C$ shifts right; negative $C$ shifts left.

• $y = \sin(x - \frac{\pi}{4})$ shifts the sine wave $\frac{\pi}{4}$ units to the right

Vertical Shift ($D$)

The vertical shift moves the entire wave up or down. The center line becomes $y = D$ instead of $y = 0$.

• $y = \sin(x) + 2$ shifts the wave up 2 units — it oscillates between 1 and 3

Worked Examples

Example 1: Identify the Properties

For $y = 4\sin(3x)$, find the amplitude and period.

• Amplitude: $|A| = |4| = 4$
• Period: $\frac{2\pi}{|B|} = \frac{2\pi}{3} \approx 2.094$

Answer: Amplitude is 4, period is $\frac{2\pi}{3}$.

Example 2: Write the Equation from a Description

An AC voltage signal oscillates between $-170$ and $170$ volts and completes 60 cycles per second. Write the equation.

• Amplitude: $A = 170$ (half the peak-to-peak distance)
• Frequency: 60 Hz means the period is $\frac{1}{60}$ second, so $B = 2\pi \times 60 = 120\pi$

$V(t) = 170\sin(120\pi t)$

Answer: $V(t) = 170\sin(120\pi t)$. This is the standard equation for 120V AC power (170V peak).

Example 3: Phase-Shifted Cosine

Describe the graph of $y = 2\cos\!\left(x - \frac{\pi}{3}\right) + 1$.

• Amplitude: 2
• Period: $\frac{2\pi}{1} = 2\pi$
• Phase shift: $\frac{\pi}{3}$ to the right
• Vertical shift: up 1

The wave oscillates between $1 - 2 = -1$ and $1 + 2 = 3$, centered on $y = 1$.

Answer: A cosine wave with amplitude 2, period $2\pi$, shifted right by $\frac{\pi}{3}$ and up by 1.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• The sine graph starts at 0 and the cosine graph starts at 1 — otherwise they have the same shape
• Amplitude ($|A|$) controls the wave height; period ($\frac{2\pi}{|B|}$) controls how wide one cycle is
• Phase shift ($C$) moves the wave left or right; vertical shift ($D$) moves it up or down
• The general form is $y = A\sin(B(x - C)) + D$ or $y = A\cos(B(x - C)) + D$
• AC electrical signals are modeled as sine waves — understanding amplitude and frequency is essential for electricians

To learn how to fit sine and cosine functions to real-world data like tides, temperature cycles, and daylight hours, see Sinusoidal Modeling.

Return to Trigonometry for more topics in this section.