Problem 53 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Draw the graphs of $$ f(x)=\sin x+\cos x $$

Step-by-Step Solution

Answer

The graph of \(f(x)=\sin(x) + \cos(x)\) is continuous and periodic. It crosses the x-axis at \(x = \frac{3\pi}{4}\) and \(x = \frac{7\pi}{4}\), reaches a peak at \(x=\frac{\pi}{4}\), and a minimum at \(x=\frac{5\pi}{4}\).

Step 1: Understand the individual functions

First, consider the function \(\sin(x)\) and \(\cos(x)\) separately. Both are periodic functions with a period of \(2\pi\). They both oscillate between -1 and 1. However, \(\sin(x)\) starts at 0 when \(x=0\), while \(\cos(x)\) starts at 1.

Step 2: Add the functions together

When combined into the function \(f(x)=\sin(x) + \cos(x)\), the new function takes the sum of the y-values of the two functions at each x-value. This changes the amplitude and phase of the wave, but it is still a periodic function.

Step 3: Graph the function

Plot the function \(f(x) = \sin(x) + \cos(x)\) on a graph. For each x-value, calculate the y-value as the sum of the y-values of \(\sin(x)\) and \(\cos(x)\) at that x-value and plot the point. Do this for a complete cycle from \(x = 0\) to \(x = 2\pi\) to see one complete cycle of the wave. Due to the periodicity, the graph repeats this pattern for larger or smaller x-values. It crosses the x-axis at \(x = \frac{3\pi}{4}\) and \(x = \frac{7\pi}{4}\), reaches a peak at \(x=\frac{\pi}{4}\), and reaches a minimum at \(x=\frac{5\pi}{4}\).