Problem 15 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

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

Step-by-Step Solution

Answer

The graph of \(f(x)=\sin x+|\sin x|\) looks just like \(y=|\sin(x)|\), but it's twice as tall when \(\sin(x)\) is positive, from 0 to \(2\pi\). All values which would be under the x-axis are elevated to zero because of the absolute value function.

Step 1: Graph of Standard Sine Function

Firstly, graph the standard sine function, \(y=\sin(x)\), on a graph. This function has a period of \(2\pi\), range from -1 to 1, crosses the x-axis at \(0, \pm \pi, \pm 2\pi, \pm3\pi,\) etc.

Step 2: Graph of Absolute Value Sine Function

Then, graph the absolute value sine function, \(y=|\sin(x)|\). This function looks similar to the standard sine function but doesn't dip below the x-axis because absolute values make all output zero or positive. It has a period of \(\pi\) instead of \(2\pi\) in the standard sine function.

Step 3: Sum Up Both Functions

Overlay the standard sine function and the absolute sine function on the same graph and visually add the corresponding y-values together from both functions to create the new function \(f(x)=\sin(x)+|\sin(x)|\). The new graph increases when \(y=\sin(x)\) is positive, but stays at zero when \(y=\sin(x)\) is negative.

Key Concepts

Sine Function

The sine function, represented as \(y = \sin(x)\), is one of the fundamental trigonometric functions. It is widely used in mathematics, engineering, and physical sciences to model periodic phenomena. The graph of the sine function is a smooth, continuous wave that oscillates above and below the x-axis.Key characteristics of the sine wave include:

• Amplitude: The peak of the wave, measured from the centerline to the crest, is 1.
• Period: The period of the sine function is \(2\pi\). This means the wave pattern repeats every \(2\pi\) units along the x-axis.
• Zeros: The function crosses the x-axis at 0, \(\pm \pi, \pm 2\pi, \pm 3\pi\), and so on.
• Range: The range is from -1 to 1, capturing the complete amplitude of the wave.

The graph of \(\sin(x)\) smoothly transitions through these points, rising and falling in a familiar undulating pattern. The sine wave is an excellent example of how trigonometric functions are used to represent cycles and periodic changes over time.

Absolute Value Function

The absolute value function is a mathematical function defined for real numbers. It outputs the non-negative value of any given input. When it applies to sine, forming the function \(y = |\sin(x)|\), it mirrors or reflects all negative portions of the sine wave to be positive. Key characteristics of the absolute value sine function include:

• Non-negative outputs: Unlike the standard sine function, which can fluctuate between -1 and 1, the absolute value sine function stays between 0 and 1. It never dips below the x-axis.
• Period: The period of \(|\sin(x)|\) is \(\pi\), not \(2\pi\). This is because the reflective symmetry with the x-axis effectively halves the repeating pattern length.
• Shape: The graph resembles a series of hills and valleys without any negative troughs.

The transformation to the sine wave due to the absolute value function allows every negative peak of the sine function to mirror about the x-axis, creating a wave that is entirely non-negative. This property is beneficial in contexts where negative values are not applicable or useful.

Periodicity in Trigonometry

Periodicity refers to the repeating nature of some functions. In trigonometry, many functions, including sine and cosine, exhibit periodic behavior. Understanding periodicity is crucial for graphing trigonometric functions as it helps in predicting and analyzing function values.For the functions \(\sin(x)\) and \(|\sin(x)|\):

• Sine's Periodicity: The sine function repeats every \(2\pi\) radians. This periodicity means that the wave-like pattern will recur indefinitely at this interval.
• Absolute Sine's Periodicity: The absolute value of the sine function shortens this repeat cycle to \(\pi\) because of the symmetry introduced by reflecting the negative values.

Periodicity in these functions is essential for determining the rhythm of the waves they describe. When these two functions are combined, as in \(f(x)=\sin(x) + |\sin(x)|\), understanding the periodic behavior helps predict the function's graph, showing exactly where it will rise or fall to zero, aiding in comprehension of both the mathematical properties and potential applications.