Determine period of a sinusoidal function

Period is the smallest positive horizontal interval over which a sinusoidal function repeats, determined from $y=a\sin(bx+c)+d$ or $y=a\cos(bx+c)+d$ by $T=\frac{2\pi}{|b|}$ in radians, or $T=\frac{360^\circ}{|b|}$ in degrees. The period controls horizontal frequency and is independent of amplitude, vertical shift, and phase shift; this treatment is limited to standard sine and cosine models rather than more advanced periodic functions or generalized frequency analysis.

Detailed Explanation: Determine period of a sinusoidal function

The period of a sinusoidal function is the smallest positive horizontal interval after which the graph repeats.

For a function in either form

y=a\sin(bx+c)+d

y=a\cos(bx+c)+d,

the period in radians is

T=\frac{2\pi}{ \vert b \vert }.

Only the coefficient $b$ affects the period. The values $a$ , $c$ , and $d$ change the height, horizontal shift, or vertical shift, but not the period.

Example

Determine the period of

y=-3\cos\left(4x-\frac{\pi}{2}\right)+2.

Step 1: Identify $b$ .

Compare the function with

y=a\cos(bx+c)+d.

The coefficient of $x$ is

b=4.

Step 2: Use the period formula.

Since the angle is in radians,

T=\frac{2\pi}{ \vert b \vert }.

Substitute $b=4$ :

T=\frac{2\pi}{ \vert 4 \vert } =\frac{2\pi}{4} =\frac{\pi}{2}.

Therefore, the period is

\boxed{\frac{\pi}{2}}.

The graph repeats every $\frac{\pi}{2}$ units horizontally.

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Sinusoidal Function Parameters (2 Params) - Function to Parameters

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