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Sinusoid

Sinusoid

A wave shaped like the graph of y = sin x. Note: The graphs of both sine and cosine are sinusoids.

 

A sinusoidal wave showing approximately 4 complete smooth, repeating S-shaped cycles of equal amplitude and period.

 

 

See also

Amplitude, period of a periodic function

Key Formula

y=Asin⁡(Bx+C)+Dy = A\sin(Bx + C) + D
Where:
  • AA = Amplitude — the height from the midline to a peak. The wave oscillates between D − |A| and D + |A|.
  • BB = Frequency factor — determines how quickly the wave repeats. The period equals 2π / |B|.
  • CC = Phase shift parameter — the horizontal shift is −C/B (positive means shift left when C > 0).
  • DD = Vertical shift — moves the midline of the wave up (D > 0) or down (D < 0).

Worked Example

Problem: Find the amplitude, period, phase shift, and vertical shift of the sinusoid y = 3 sin(2x − π) + 1, then state its maximum and minimum values.
Step 1: Identify A, B, C, and D by comparing with the general form y = A sin(Bx + C) + D.
A=3,B=2,C=−π,D=1A = 3,\quad B = 2,\quad C = -\pi,\quad D = 1
Step 2: Compute the amplitude. The amplitude is the absolute value of A.
Amplitude=∣A∣=∣3∣=3\text{Amplitude} = |A| = |3| = 3
Step 3: Compute the period using the formula 2π / |B|.
Period=2π∣2∣=π\text{Period} = \frac{2\pi}{|2|} = \pi
Step 4: Compute the phase shift using −C/B. A positive result means the graph shifts to the right.
Phase shift=−CB=−−π2=π2 (right)\text{Phase shift} = -\frac{C}{B} = -\frac{-\pi}{2} = \frac{\pi}{2} \text{ (right)}
Step 5: The vertical shift is D = 1, so the midline is y = 1. The maximum value is D + |A| and the minimum is D − |A|.
Max=1+3=4,Min=1−3=−2\text{Max} = 1 + 3 = 4,\qquad \text{Min} = 1 - 3 = -2
Answer: Amplitude = 3, period = π, phase shift = π/2 to the right, vertical shift = 1 (midline y = 1), maximum value = 4, minimum value = −2.

Another Example

This example works in reverse: instead of reading parameters from a given equation, you build the equation from a set of given properties. It also uses cosine rather than sine to reinforce that both produce sinusoids.

Problem: Write the equation of a sinusoid that has amplitude 5, period 4π, is shifted π/3 to the left, and has a midline at y = −2. Express it using cosine.
Step 1: Start with the general cosine form y = A cos(Bx + C) + D. Both sine and cosine produce sinusoids, so either form works.
y=Acos⁡(Bx+C)+Dy = A\cos(Bx + C) + D
Step 2: The amplitude is 5, so A = 5. The midline is y = −2, so D = −2.
A=5,D=−2A = 5,\quad D = -2
Step 3: Find B from the period. Set 2π / |B| = 4π and solve.
∣B∣=2π4π=12,so B=12|B| = \frac{2\pi}{4\pi} = \frac{1}{2},\quad\text{so } B = \frac{1}{2}
Step 4: A shift of π/3 to the left means phase shift = −π/3. Since phase shift = −C/B, solve for C.
−C1/2=−π3  ⟹  C=π6-\frac{C}{1/2} = -\frac{\pi}{3} \implies C = \frac{\pi}{6}
Step 5: Substitute all values into the equation.
y=5cos⁡ ⁣(12x+π6)−2y = 5\cos\!\left(\tfrac{1}{2}x + \tfrac{\pi}{6}\right) - 2
Answer: y = 5 cos(x/2 + π/6) − 2

Frequently Asked Questions

Is a cosine graph a sinusoid?
Yes. A cosine curve is a sinusoid because cos x = sin(x + π/2). The cosine graph is just the sine graph shifted π/2 to the left. Any function of the form A cos(Bx + C) + D can be rewritten as an equivalent sine function, so both are sinusoids.
What is the difference between a sinusoid and a sine wave?
The terms are often used interchangeably, but strictly speaking a "sine wave" refers to the basic y = sin x shape, while "sinusoid" is the broader family that includes any amplitude, period, phase shift, or vertical shift applied to that shape. Every sine wave is a sinusoid, but not every sinusoid is the plain y = sin x curve.
How do you find the equation of a sinusoid from its graph?
Read four pieces of information from the graph: the midline (D), the distance from midline to peak (amplitude A), the horizontal length of one full cycle (period, which gives B = 2π / period), and where the first peak or zero-crossing occurs (which determines the phase shift C). Plug these into y = A sin(Bx + C) + D.

Sinusoid (sine form) vs. Sinusoid (cosine form)

Sinusoid (sine form)Sinusoid (cosine form)
General equationy = A sin(Bx + C) + Dy = A cos(Bx + C) + D
Value at x = 0 (basic form)Starts at the midline (y = 0)Starts at the peak (y = 1)
Relationshipsin x = cos(x − π/2)cos x = sin(x + π/2)
When to useConvenient when the wave crosses the midline at x = 0Convenient when the wave is at a maximum or minimum at x = 0

Why It Matters

Sinusoids appear throughout science and engineering: sound waves, alternating current in electrical circuits, tides, and seasonal temperature data are all modeled by sinusoidal functions. In precalculus and trigonometry courses, you need to identify, graph, and write equations for sinusoids as a foundational skill. Understanding sinusoids also prepares you for Fourier analysis, where complex periodic signals are broken down into sums of sinusoids.

Common Mistakes

Mistake: Confusing the phase-shift sign: assuming C > 0 shifts the graph to the right.
Correction: In y = A sin(Bx + C) + D, the phase shift is −C/B. When C is positive, the graph shifts to the left, not the right. Factor the argument as B(x + C/B) to see the direction clearly.
Mistake: Using 2π / A instead of 2π / |B| for the period.
Correction: The period depends only on B (the coefficient of x), not on A (the amplitude). A controls the height of the wave; B controls how quickly it repeats.

Related Terms