Problem 33 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Draw the graphs of $$ f(x)=\cos \left(\frac{x}{3}\right) $$

Step-by-Step Solution

Answer

The graph of the function \( f(x) = \cos \left( \frac{x}{3} \right) \) is a wave with an amplitude of 1 and a period of \( 6\pi \) which passes through the points \( (0, 1), (3\pi, -1), (6\pi, 1) \)

Step 1: Identify the amplitude

The amplitude of the cosine function, \( f(x) = \cos \left( \frac{x}{3} \right) \), is 1. This is the maximum distance the function can move from the center line (usually the x-axis).

Step 2: Identify the period

The period of the function, \( f(x) = \cos \left( \frac{x}{3} \right) \), is \( 2\pi*3 = 6\pi \). This is the distance between identical points on the wave. In this case, the wave repeats every \( 6\pi \) units.

Step 3: Plotting the critical points

For the function, \( f(x) = \cos \left( \frac{x}{3} \right) \), let us plot the points for a full period. To plot, choose points over a range of \( 0 \) to \( 6\pi \) as it is the function period. The critical points occur at approximated locations of \( x = 0, 3\pi, 6\pi \) where function values are \( f(0) = 1, f(3\pi) = -1, f(6\pi) = 1 \). (Since \( \cos(0) = 1, \cos(\pi) = -1, \cos(2\pi) = 1 \)). Write these points on the graph.

Step 4: Draw the graph

Connect the plotted points with a smooth wave that shows the periodic and oscillating behavior of the function \( f(x) = \cos \left( \frac{x}{3} \right) \). Make sure the wave goes through each plotted point and accurately represents the up and down behavior of the cosine function.

Key Concepts

Amplitude of Cosine Function

In the world of trigonometric functions, the amplitude speaks to how tall or short a wave can be. Specifically with the cosine function, the amplitude represents the maximum value it reaches from its center position, which we typically think of as the horizontal axis in a graph. For the standard cosine function, which is \( f(x) = \text{cos}(x) \), the amplitude is 1. This means the graph rises and falls 1 unit above and below the center line.

In the exercise, we’re given the function \( f(x) = \text{cos} \left(\frac{x}{3}\right) \). Although it might seem trickier because of the fraction inside the cosine, the function's amplitude remains unaffected and is still 1. This means no matter what \( x \) value you choose, the function will never exceed 1 or drop below -1. When you prepare to graph this function, remember to mark up and down from the center line by one unit to truly capture the amplitude of your wave.

Period of Trigonometric Function

As we journey through the cycles of a trigonometric function, we come across the concept of the period. This is the horizontal length over which the function completes one full cycle of its pattern. A standard cosine function, \( f(x) = \text{cos}(x) \), goes through its peaks and valleys and back to the starting point over an interval of \( 2\pi \) units.

For the function \( f(x) = \text{cos} \left(\frac{x}{3}\right)\), we're dealing with a stretched-out version of the standard cosine wave. To find the period, we multiply \( 2\pi \) by the number at the denominator of the fraction inside the cosine function. The result is \( 6\pi \), which is the new period of the function. This stretching means that the wave is more drawn out along the x-axis, taking more horizontal space to complete its cycle. Knowing this will guide you in sketching the graph, ensuring you spread the peaks and troughs \( 6\pi \) units apart.

Plotting Critical Points

To accurately represent trigonometric functions on a graph, pinpointing critical points is a must. These are the points where the function reaches its maximum and minimum values (peaks and valleys) and crosses the center line (zeroes).

With our function, \( f(x) = \text{cos} \left(\frac{x}{3}\right) \), these points occur at intervals along the period we just calculated. For one full cycle of \( 6\pi \) units, you will spot them at \( x = 0, 3\pi, \text{ and } 6\pi \), with the function obtaining values of \( f(0) = 1, f(3\pi) = -1, \text{ and } f(6\pi) = 1 \). Placing these critical points on your graph is like setting the foundation for your wave. Once you connect these points with a smooth curve, mirroring the characteristic shape of the cosine function, your graph will beautifully display the function's oscillating nature.