Problem 37 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Draw the graphs of $$ f(x)=\cos \left(x+\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

The graph of \(y=\cos \left(x + \frac{\pi}{4}\right)\) is the same as the graph of \(y=\cos(x)\), but shifted \(\frac{\pi}{4}\) units to the left along the x-axis.

Step 1: Understand the basic cosine function

Firstly, it's essential to understand the standard cosine function, \(y=\cos(x)\), before diving into any variations. The function \(y=\cos(x)\) is a periodic function that oscillates between -1 and 1 and has a period of \(2\pi\). This function starts at a maximum point at x=0, descends to a minimum at \(x=\pi\), and then ascends back to a maximum at \(x=2\pi\). This is one full period of the function.

Step 2: Understand the phase shift

Next, notice that the function in this problem isn't simply \(y=\cos(x)\), but instead \(y=\cos \left(x + \frac{\pi}{4}\right)\). This means that the cosine function is shifted by \(-\frac{\pi}{4}\) units to the left. This is called a phase shift. The function still oscillates between -1 and 1 and has a period of \(2\pi\). However, each point on this new graph set of points \(\left(x, \cos \left(x + \frac{\pi}{4}\right)\right)\) corresponds to the points \(\left(x + \frac{\pi}{4}, \cos(x)\right)\) on the standard cosine graph.

Step 3: Plot the function

Having the above information, draw a cosine curve starting at \(\left(-\frac{\pi}{4}, 1\right)\), reaching a minimum at \(\frac{3\pi}{4}\), then going back up to complete the cycle at \(\frac{9\pi}{4}\). Examine the pattern of the cosine graph and continue it along the x-axis.

Key Concepts

Phase Shift in Trigonometry

Phase shift is a key concept when dealing with trigonometric functions like sine and cosine. It refers to the horizontal translation of a graph along the x-axis. This means moving the entire graph left or right.

For the function \(f(x)=\cos \left(x+\frac{\pi}{4}\right)\), the expression inside the cosine, \(x + \frac{\pi}{4}\), indicates a phase shift. This is because the argument of the cosine function is modified by some constant. To identify the direction of the shift:

• Addition inside the cosine function, like \(x + \frac{\pi}{4}\), results in a shift to the left.
• Subtraction would result in a shift to the right.

In our example, the graph of the cosine function shifts \(-\frac{\pi}{4}\) units to the left compared to the basic cosine function, \(y = \cos(x)\).

This phase shift changes where the cycle of the waveform starts and ends, but it does not affect the height (amplitude) or the length (period) of one complete cycle.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or cycles. Both sine and cosine functions are perfect examples of this due to their repeating patterns over a fixed period.

The cosine function, \(y=\cos(x)\), is periodic with a period of \(2\pi\). This means after every \(2\pi\) units along the x-axis, the function starts a new cycle.

• The period is the length of one full cycle on the graph.
• For \(y=\cos(x)\), the pattern from the peak at 1 to another peak is \(2\pi\), moving through zeros and a minimum along the way.

In our shifted function, \(y=\cos \left(x+\frac{\pi}{4}\right)\), the period remains \(2\pi\), maintaining the oscillation properties of the cosine function. Thus, despite the shift in the starting point due to the phase shift, the frequency of repeating cycles remains unchanged.

Trigonometric Graph Transformations

Graph transformations in trigonometry involve modifying basic trigonometric functions to obtain new ones. The cosine graph, like other trigonometric graphs, can undergo transformations such as phase shifts, vertical shifts, stretches, shrinks, and reflections.

The function \(f(x)=\cos \left(x+\frac{\pi}{4}\right)\) demonstrates a phase shift transformation. Let's explore common transformations:

• Phase Shifts: Horizontal shifts like in our function, \(\cos \left(x+\frac{\pi}{4}\right)\), where the addition moves it to the left.
• Amplitude Changes: Multiplying the function by a constant changes the height of the wave; e.g., \(2\cos(x)\) doubles the height.
• Vertical Shifts: Adding a constant outside the function, such as \(\cos(x) + C\), shifts the graph up or down.
• Reflections: Negating the function \(-\cos(x)\) flips it across the x-axis.
• Stretches and Shrinks: Multiplying the angle \(x\) by a factor compresses or expands the cycle.

Understanding these transformations allows you to predict and draw various forms of trigonometric functions simply based on a function's equation.