Problem 10 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Draw the graphs of $$ f(x)=\sin (x+1) $$

Step-by-Step Solution

Answer

The graph of \(f(x) = \sin (x + 1)\) moves in the typical pattern of a sine function, but shifted one unit to the left. It starts from \(x=-1\), goes to y=1 at \(x=-1/2\), y=0 at \(x=0\), y=-1 at \(x=1/2\) and y=0 at \(x=\pi\). This pattern continues periodically.

Step 1: Understanding the Sine Function

The sine function is a periodic function with a period of \(2\pi\), meaning it repeats every \(2\pi\) units. It oscillates between -1 and 1. Typically, it starts at \(x = 0\), goes up to 1 at \(x = \pi/2\), back down to 0 at \(x = \pi\), down to -1 at \(x = 3\pi/2\), and back to 0 at \(x = 2\pi\). Now, this movement will be shifted to the right by 1 unit due to the term \(x+1\) in the function.

Step 2: Applying the Shift

The term \(x+1\) inside the function indicates a shift to the left by 1 unit. So, instead of starting at \(x=0\), this function will start from \(x = -1\). So, \(f(-1) = 0\). This should be marked as the starting point.

Step 3: Draw the Sine Function

From the starting point, follow the typical path of the sine function: up to 1 at \(x = -1 + \pi/2 = -1/2\), back down to 0 at \(x = -1 + \pi = 2\), down to -1 at \(x = -1 + 3\pi/2 = 1/2\), and back to 0 at \(x = -1 + 2\pi = \pi\). Repeat this pattern as needed to graph the function.

Key Concepts

sine function

The sine function is a fundamental concept in trigonometry and waves. It's crucial to understand that the sine function, denoted as \( \sin(x) \), represents how an angle in a unit circle maps to the y-coordinate. Hence, it oscillates between -1 and 1 as the angle x changes.

The shape of the sine wave is smooth and periodic. This means that it repeats itself in a regular cycle. It starts from zero at \( x = 0 \), peaks at 1 at \( x = \frac{\pi}{2} \), descends back to zero at \( x = \pi \), valleys at -1 at \( x = \frac{3\pi}{2} \), and returns to zero at \( x = 2\pi \). By repeating this pattern, the function extends indefinitely in both directions on the x-axis.

• The amplitude of the sine function is 1, which tells us the height from the middle of the wave to the peak.
• It is symmetric about the origin, meaning the wave looks the same on both sides of the y-axis.
• The sine wave is known for its smooth and continuous flow.

periodic function

Periodic functions are essential in understanding patterns that repeat over time. A function is labeled periodic if it reoccurs at regular intervals, or 'periods', as it moves along the x-axis. The sine function is a classic example, with a period of \(2\pi\). This indicates that every \(2\pi\) units, the wave pattern repeats itself.

In mathematics, periodicity is a critical concept because it helps simplify and identify predictable parts of complicated functions or datasets.

• Periodic functions are integral to modeling phenomena such as sound waves, tides, and seasons.
• A key property is that if \( f(x) \) is periodic with period \( T \), then \( f(x) = f(x + T) \) for all values of x.
• For the sine function, every full period (\(2\pi\)) is identical in shape and structure to the one before it.

phase shift

Phase shift is a transformation applied to periodic functions that results in a horizontal shift along the x-axis. It is crucial for altering the starting point of a function without changing its shape.

In the context of the sine function, phase shift is determined by a change inside the angle. For example, \( \sin(x + 1) \) has a horizontal shift. In this case, the '+1' indicates the curve shifts to the left by 1 unit.

• Positive values inside result in a shift left.
• Negative values lead to a shift right.
• Phase shift doesn’t alter the amplitude or the period, but simply repositions the function’s graph horizontally.

By mastering the concept of phase shift, understanding graphical transformations on the x-axis becomes intuitive.