Question

Determine the period, in radians, of each function using two different methods. a) \(y=-2 \sin 3 x\) b) \(y=-\frac{2}{3} \cos \frac{\pi}{6} x\)

Short answer

The period of \( y = -2 \sin 3x \) is \( \frac{2\pi}{3} \) radians. The period of \( y = -\frac{2}{3} \cos \frac{\pi}{6}x \) is 12 radians.

Step-by-step solution

Identify the standard form for trigonometric functions

The period of a trigonometric function is determined from its standard form. For sine and cosine functions, the standard form is given by: For sine: \( y = a \sin (bx + c) + d \) For cosine: \( y = a \cos (bx + c) + d \) Here, the period is calculated using the coefficient of \(x\), which is \(b\) in the expression.

Determine the period of \( y = -2 \sin 3x \) using the coefficient of x

In the given function \( y = -2 \sin 3x \), the coefficient of \(x\) is 3.The period \(T\) of the sine function is given by: \[ T = \frac{2\pi}{b} \] Substituting \(b = 3\): \[ T = \frac{2\pi}{3} \]

Determine the period of \( y = -\frac{2}{3} \cos \frac{\pi}{6}x \) using the coefficient of x

In the given function \( y = -\frac{2}{3} \cos \frac{\pi}{6}x \), the coefficient of \(x\) is \(\frac{\pi}{6}\).The period \(T\) of the cosine function is given by: \[ T = \frac{2\pi}{b} \] Substituting \( b = \frac{\pi}{6} \): \[ T = \frac{2\pi}{\frac{\pi}{6}} = \frac{2\pi \times 6}{\pi} = 12 \]

Method 1: Verify periods by graph analysis

For \( y = -2 \sin 3x \), graph the function on a calculator or graphing software and confirm that one full cycle completes in \( \frac{2\pi}{3} \) radians.For \( y = -\frac{2}{3} \cos \frac{\pi}{6}x \), graph the function on a calculator or graphing software and confirm that one full cycle completes in 12 radians.

Key concepts

sine function period

The sine function is one of the fundamental trigonometric functions. Its standard form is given by
\( y = a \sin (bx + c) + d \)
Here, 'a' affects the amplitude, 'b' affects the period, 'c' affects the phase shift, and 'd' affects the vertical shift. The period of the sine function is influenced by the coefficient of x, 'b'. To determine the period (T) of a sine function, use the formula:
\( T = \frac{2\pi}{b} \)

For example, consider the function \( y = -2 \sin 3x \). Here, 'b' is 3. Substituting 'b' into the period formula:
\( T = \frac{2\pi}{3} \). This means the sine function completes one full cycle every \( \frac{2\pi}{3} \) radians.

To verify, you can graph the function and observe that one complete wave from peak to peak indeed covers \( \frac{2\pi}{3} \) radians. This understanding can be applied to any sine function to ascertain its period based on the coefficient of 'x'.

cosine function period

The cosine function shares many characteristics with the sine function. Its standard form is:
\( y = a \cos (bx + c) + d \)

Just as in the sine function, 'a' affects amplitude, 'b' affects the period, 'c' affects the phase shift, and 'd' affects the vertical shift. The formula for the period (T) of the cosine function is:
\( T = \frac{2\pi}{b} \)

To illustrate, let’s use the function \( y = -\frac{2}{3} \cos \frac{\pi}{6}x \). Here, 'b' is \frac{\pi}{6}. Substituting 'b' into the period formula:
\( T = \frac{2\pi}{\frac{\pi}{6}} = \frac{2\pi \times 6}{\pi} = 12 \). Therefore, the cosine function completes one full cycle every 12 radians.

To confirm this, you can draw the graph and see that it completes one entire cycle over an interval of 12 radians. Recognizing this pattern helps in identifying the period of any cosine function based on the coefficient of 'x'.

trigonometric functions

Trigonometric functions are essential in mathematics, especially in describing waves and periodic phenomena. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). Each function has a periodic nature, meaning they repeat at regular intervals.
For sine and cosine functions, these intervals (or periods) can be determined using the coefficient of 'x' in their respective standard forms:
\( y = a \sin (bx + c) + d \)
\( y = a \cos (bx + c) + d \)

Their period is given by:
\( T = \frac{2\pi}{b} \)

Understanding the periods of these functions is crucial in graphing them correctly and analyzing real-world phenomena, such as sound waves, light waves, and tides. By altering the coefficient 'b', you can stretch or compress the graph of the function horizontally, changing its cycle length. This knowledge is foundational for many advanced topics in mathematics and physics.
Using graphing tools to visualize these functions helps solidify the concept of periodicity and the impact of the coefficient 'b'. Practice plotting different trigonometric functions to see how changes in 'b', 'a', 'c', and 'd' transform the graph.