Amplitude Formula for Sine and Cosine Waves

Key Points

• The amplitude formula is $\lvert A \rvert$, the absolute value of the coefficient in front of the sine or cosine.

• Equivalently, amplitude is $\frac{\text{max}-\text{min}}{2}$, half the vertical distance between the highest and lowest points.

• Amplitude is always positive, so a negative coefficient such as $-4$ still gives an amplitude of 4 while flipping the wave upside down.

• The vertical shift $D$ moves the whole wave but never changes its amplitude, since both the maximum and minimum move together.

• Amplitude measures height while period measures width, so the coefficient $B$ affects the period and not the amplitude.

What Is the Amplitude Formula?

For a sinusoid written as $y = A\sin(Bx+C)+D$, the amplitude formula is:

$$\text{amplitude} = \lvert A \rvert$$

When the equation is not available but the graph or a data table is, use the equivalent form:

$$\text{amplitude} = \frac{y_{\max} - y_{\min}}{2}$$

Why Amplitude Is Half the Range, Not the Whole Range

A plain sine wave runs from $-1$ up to $+1$. The total vertical travel is 2, but the amplitude is 1, because amplitude measures the distance from the centre line to a peak rather than from trough to peak.

That centre line sits at the average of the maximum and minimum, which is the vertical shift $D$. Measuring from there to the top gives exactly half the full range.

Halving is also what makes the vertical shift irrelevant. Adding 5 to every value raises the maximum and the minimum by 5 each, so their difference is unchanged and the amplitude stays the same.

Variable Key

Examples of the Amplitude Formula

Example 1

Find the amplitude of $y = 5\sin x$.

$$\lvert A \rvert = \lvert 5 \rvert = 5$$

Final answer: 5

Example 2

Find the amplitude of $y = -4\cos x$ by reading the coefficient directly.

The coefficient is $-4$, so reporting an amplitude of $-4$ looks like the obvious reading.

An amplitude is a distance, and a distance cannot be negative. The minus sign reflects the wave vertically, swapping its peaks and troughs, but the wave still reaches 4 units either side of the centre line.

Taking the absolute value:

$$\lvert -4 \rvert = 4$$

Final answer: 4. The negative sign changes the wave's orientation, not its height.

Example 3

Find the amplitude of $y = 3\sin(2x) + 7$.

The amplitude depends only on $A$:

$$\lvert A \rvert = 3$$

Final answer: 3

The 2 inside the bracket halves the period, and the $+7$ lifts the centre line to $y=7$, but neither touches the amplitude.

Example 4

A wave has maximum 11 and minimum 3. Find its amplitude and centre line.

$$\text{amplitude} = \frac{11-3}{2} = 4, \qquad D = \frac{11+3}{2} = 7$$

Final answer: amplitude 4, centre line $y = 7$.

Half the difference gives the amplitude and half the sum gives the centre, so one pair of readings supplies both.

Example 5

A tide varies between 0.4 m and 4.6 m. Write a sine model and state its amplitude.

$$\text{amplitude} = \frac{4.6-0.4}{2} = 2.1, \qquad D = \frac{4.6+0.4}{2} = 2.5$$

Final answer: amplitude 2.1 m, with a model of the form $y = 2.1\sin(Bx+C)+2.5$.

The value of $B$ would come from the tidal period, which is a separate measurement from the amplitude.

Don't Confuse Amplitude With These Three

Amplitude and period are the pair most often swapped. One is a height and the other a width, and they are controlled by different coefficients, $A$ and $B$. The periodic function page covers periodicity itself.

Where Do Students Trip Up on Amplitude?

Mistake 1: Reporting a negative amplitude

Where it slips in: The coefficient carries a sign and gets copied straight across.

Don't do this: Amplitude $= -4$ for $y = -4\cos x$.

The correct way: Amplitude $= \lvert -4 \rvert = 4$. A negative coefficient flips the wave but cannot give it negative height.

Students who ask whether the answer is a distance before writing it down never report a negative amplitude, because distances have no sign. The same question settles magnitudes, lengths and absolute values throughout the subject.

Mistake 2: Using the full range instead of half

Where it slips in: The gap between maximum and minimum is the visible feature on a graph.

Don't do this: Amplitude $= y_{\max} - y_{\min}$

The correct way: Halve it. For a maximum of 11 and a minimum of 3 the amplitude is 4, not 8.

Mistake 3: Letting the period affect the amplitude

Where it slips in: Both are properties of the same wave, and both come from coefficients in the same equation.

Don't do this: Change the amplitude when $B$ changes.

The correct way: $A$ controls height and $B$ controls width. In $y = 3\sin(2x)$ the amplitude is 3 regardless of the 2, which only compresses the wave horizontally.

Key Takeaways

• The amplitude formula is $\lvert A \rvert$, or $\frac{y_{\max}-y_{\min}}{2}$ from a graph or data.

• Amplitude is a distance, so it is never negative however the coefficient is signed.

• The vertical shift and the period coefficient leave the amplitude untouched.

• Half the difference of the extremes gives the amplitude, and half their sum gives the centre line.

• To work through trigonometric graphs with a teacher, explore a trigonometry tutor or math classes online.

Practice These Three Before Moving On

Find the amplitude of $y = -7\sin(3x)+1$. Then find the amplitude and centre line of a wave running between $-2$ and 14, and say what happens to each if every value is increased by 5.

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