Clear doubts with RD Sharma Class 11 Solutions Chapter 6 - Graphs of Trigonometric Functions (Ex 6.1) Exercise 6.1 - Free PDF

Studying the graphs in RD Sharma Chapter 6 visually clarifies several core properties of trigonometric functions. Key characteristics you can observe are:

• Domain and Range: The horizontal and vertical extent of the graph directly shows the function's valid inputs (domain) and outputs (range).

• Periodicity: You can see the specific interval after which the graph pattern repeats itself, which defines the period of the function (e.g., 2π for sin(x) and cos(x), and π for tan(x)).

• Symmetry: The graphs reveal whether a function is even (like cos(x), symmetric about the y-axis) or odd (like sin(x), symmetric about the origin).

• Asymptotes: For functions like tan(x), sec(x), and cosec(x), the graphs clearly illustrate the vertical lines where the function is undefined.

The solutions for RD Sharma Chapter 6 explain the method for plotting y = sin(x) as follows:

• First, create a table of values for x and sin(x) at standard angles like 0, π/6, π/2, π, 3π/2, and 2π.

• Next, mark these (x, y) coordinates on the graph paper.

• Draw a smooth, continuous curve passing through these points, ensuring the shape is a wave.

• Finally, indicate that the domain is all real numbers (ℝ) and the range is [-1, 1]. The graph repeats every 2π interval, which is its period.

The RD Sharma solutions explain graph transformations in a systematic way. To solve a problem like plotting y = 3 + cos(x), the method involves:

• Step 1: Start with the basic graph of y = cos(x), which oscillates between -1 and 1.

• Step 2: Apply the vertical shift. The '+3' indicates that the entire graph of cos(x) must be shifted vertically upwards by 3 units.

• Step 3: The new range of the function will be [-1+3, 1+3], which is [2, 4]. The solutions visually demonstrate this shift, helping students understand how constants modify the base graph.

The period of a function is the length of the smallest interval over which its graph repeats. The RD Sharma graphical solutions make the difference clear:

• The graph of sin(x) completes one full cycle (one crest and one trough) over an interval of 2π. After this, the wave pattern repeats.

• The graph of tan(x) consists of repeating branches separated by vertical asymptotes. One complete branch of the curve repeats every π units. For instance, the shape between x = -π/2 and x = π/2 is identical to the shape between x = π/2 and x = 3π/2.

The solved examples visually confirm that sin(x) requires a wider interval (2π) to repeat its full pattern compared to tan(x) (π).

A common mistake when plotting y = sec(x) is forgetting to draw the vertical asymptotes. Since sec(x) = 1/cos(x), it is undefined whenever cos(x) = 0. The RD Sharma solutions prevent this error by teaching a structured approach:

1. First, lightly sketch the graph of the reciprocal function, y = cos(x).

2. Identify the points where the cos(x) graph crosses the x-axis (e.g., at x = ±π/2, ±3π/2, etc.).

3. Draw vertical dotted lines (asymptotes) at these x-values.

4. Finally, draw the U-shaped curves of the sec(x) graph between the asymptotes, touching the peaks and troughs of the cos(x) graph. This method ensures you never miss the asymptotes.

NCERT Class 11 Maths Chapter 3 introduces the fundamental concepts of trigonometric functions, their definitions, and identities. RD Sharma's Chapter 6 builds directly on this foundation by providing an in-depth, graphical exploration. While NCERT establishes the 'what', RD Sharma solutions for this chapter focus on the 'how' and 'why' from a visual perspective, helping you understand properties like periodicity, amplitude, domain, and range by seeing them on a graph, which is essential for solving more complex problems.