Trigonometric Functions

How to find amplitude and period?

Quick answer

For an equation like y = a sin(bx) or y = a cos(bx), find the amplitude by taking the absolute value of a, |a|, and the period in radians by dividing 2π by |b| (or 360° by |b| in degrees). Amplitude measures wave height from center, while period measures the horizontal length of one complete cycle.

The topic explained

The basic rule for amplitude and period

When studying trigonometric functions, understanding these core properties lets you read a wave equation quickly. For functions written in the form y = a sin(bx) or y = a cos(bx), the amplitude is |a| and the period is 2π / |b|.

Amplitude is the vertical distance from the center line to a peak or a trough. Because distance is always positive, amplitude is never negative, even if a is a negative number. The period is the horizontal distance along the x-axis that the function takes to complete one full cycle before repeating.

Step-by-step example with numbers

Take the function y = -4 cos(3x). First, find the amplitude by reading the value of a in front of the cosine term. Here, a = -4. Take its absolute value: |-4| = 4. The amplitude is 4.

Next, find the period by reading the value of b inside the argument next to x. Here, b = 3. Divide standard full-turn radians, 2π, by |3|. The period is 2π/3. This means the cosine wave completes one full wave every 2π/3 radians on the x-axis.

Adding horizontal and vertical shifts

In a full transformed equation like y = a sin(b(x - c)) + d, shifts do not change the amplitude or the period. The vertical shift d moves the center midline up or down, while the phase shift c moves the wave left or right.

A common error is mistaking the total vertical span from trough to peak for the amplitude. Amplitude is only half of that total height. Another frequent mistake is forgetting to factor out b when a horizontal shift is written as (bx - c). Always factor b out first so the period remains 2π / |b|.

Free printable worksheet

Finding Amplitude and Period of Sine and Cosine

For 11th-grade students. State the amplitude and period for each given function. Express periods in exact radian form.

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Finding Amplitude and Period of Sine and CosineName:

Find the amplitude and period of y = 3 sin(x).
Find the amplitude and period of y = cos(4x).
Find the amplitude and period of y = -5 sin(2x).
Find the amplitude and period of y = 0.5 cos(x/3).
Find the amplitude and period of y = -2 sin(πx) + 4.
A cosine wave has a maximum value of 7 and a minimum value of -1, completing one cycle every 4 radians. Write its amplitude and period.

Common questions

Can the amplitude ever be negative?

No. Amplitude represents distance, which is always non-negative. If an equation has y = -3 sin(x), the negative sign reflects the graph across the midline, but the amplitude itself is |-3| = 3.

How do you calculate period when working in degrees?

In degrees, a standard cycle for sine and cosine is 360 degrees instead of 2π radians. To find the period, divide 360° by |b|.

Does tangent have an amplitude?

No. Tangent curves continue upward and downward toward infinity near vertical asymptotes. Because the curve has no peak or trough, amplitude is undefined for tangent functions.

How does vertical shift affect amplitude?

A vertical shift does not change amplitude at all. It shifts the midline of the curve up or down, but the vertical distance from that midline to the maximum point remains unchanged.

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