Amplitude and Period of Trigonometric Functions

Trigonometric functions like sine (sin) and cosine (cos) repeat themselves after an interval indefinitely and are thus called periodic functions. 

While analyzing the nature of these functions, we analyze their graph for properties like amplitude, period, and frequency.

The amplitude represents the height of the curve, while the period measures the length of one complete cycle of the function.

Amplitude

The amplitude of a trigonometric function refers to the height of its peaks (highest points) or the depth of its troughs (lowest points) relative to its midline, also known as the central axis or equilibrium position. 

Thus, amplitude measures how far the function oscillates above and below its central axis.

The peaks and troughs of the function oscillate symmetrically around the midline. Thus, 

• The larger amplitude leads to taller waves
• The smaller amplitude results in shorter waves

Formula

For a general trigonometric function

y = A sin x or y = A cos x,

The amplitude is |A| 

Here,

A is the coefficient of the function, and the modulus gives its positive value.

Note: For the other trigonometric functions, the amplitude does not exist.

Finding Amplitude of y = 3 sin 4x

Given the function is: y = 3 sin 4x …..(i)

Step 1: Identifying the General Form of the Function 

As we know, the general form of the function is y = A sin x …..(ii)

Step 2: Finding the Amplitude

Comparing the equations (i) and (ii), we get

The coefficient of cos x is A = 3

Thus, amplitude = |A| = |3| = 3

Finding Amplitude of y = -3 cos x

Given the function is: y = -3 cos x …..(i)

Step 1: Identifying the General Form of the Function 

As we know, the general form of the function is y = A cos x …..(ii)

Step 2: Finding the Amplitude

Comparing the equations (i) and (ii), we get

The coefficient of cos x is A = -3

Thus, amplitude = |A| = |-3| = 3

Here, the graph oscillates between -3 and 3, with a reflection about the x-axis due to the negative coefficient.

Period

The period of a trigonometric function is the length of the interval required for the function to complete one full cycle. 

For sine and cosine functions, the standard period is 2π, but it can be altered by a horizontal scaling factor.

Here,

• The closer the peaks and troughs, the shorter the period.
• Wider spacing indicates a longer period.

Formula

For a general trigonometric function

y = sin Bx or y = cos Bx

The period is ${\dfrac{\langle regular \  period\rangle }{\left| B\right| }}$

= ${\dfrac{2\pi }{\left| B\right| }}$

Here, B is the angular coefficient.

Similarly, the formula to find the period of the other 4 basic trigonometric functions is:

• Cosecant = ${\dfrac{2\pi }{\left| B\right| }}$
• Secant = ${\dfrac{2\pi }{\left| B\right| }}$
• Tangent = ${\dfrac{\pi }{\left| B\right| }}$
• Cotangent = ${\dfrac{\pi }{\left| B\right| }}$

Finding Period of y = 3 sin 4x

Given the function is y = 3 sin 4x …..(i)

Step 1: Identifying the General Form of the Function 

As we know, the general form of the function is y = A sin Bx …..(ii)

Step 2: Finding the Period

Comparing the equations (i) and (ii), we get

The angular coefficient of sin Bx is B = 4

Since the regular period of the sine function is 2π

Thus, the period is ${\dfrac{2\pi }{\left| B\right| }}$ = ${\dfrac{2\pi }{\left| 4\right| }}$ = ${\dfrac{\pi }{2}}$

Finding Period of y = -3 cos 2x

Given the function is y = -3 cos 2x …..(i)

Step 1: Identifying the General Form of the Function 

As we know, the general form of the function is y = A cos Bx …..(ii)

Step 2: Finding the Period

Comparing the equations (i) and (ii), we get

The angular coefficient of cos Bx is B = 2

Since the regular period of the cosine function is 2π

Thus, the period is ${\dfrac{2\pi }{\left| B\right| }}$ = ${\dfrac{2\pi }{\left| 2\right| }}$ = π

Solved Examples