3 Marks Question

Question 13 Marks

Sketch the graphs of the following pairs of functions on the same axes:
$\text{f(x)}=\sin2\text{x},\text{g(x)}=2\sin\text{x}$

Answer

$\text{f(x)}=\sin2\text{x},\text{g(x)}=2\sin\text{x}$

Clearly, $\sin2\text{x}$ and $2\sin\text{x}$ is a periodic function with period $\pi$ and $2\pi,$ respectively.

The graphs of $\text{f(x)}=\sin2\text{x}$ and $\text{g(x)}=2\sin\text{x}$ on different axes are shown below:

If these two graphs are drawn on the same axes, then the graph is shown below.

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Question 23 Marks

Sketch the graphs of the following trigonometric functions:
$\text{f(x)}=\cos\Big(\text{x}-\frac{\pi}{4}\Big)$

Answer

$\text{f(x)}=\cos\Big(\text{x}-\frac{\pi}{4}\Big)$
$\text{y}=\cos\Big(\text{x}-\frac{\pi}{4}\Big)$
$\Rightarrow\text{y}-0=\cos\Big(\text{x}-\frac{\pi}{4}\Big) ...(\text{i})$
On shifting the origin at $\Big(\frac{\pi}{4},0\Big),$ we get:
$\text{x = X}+\frac{\pi}{4}$ and $\text{y = Y}+0$
On subsitituting the values in (i) we get:
$\text{Y}=\cos\text{X}$
Then, we draw the graph of $\text{Y}=\cos\text{X}$ and shift it by $\frac{\pi}{4}$ to the right.
Then, we obtain the following graph:

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Question 33 Marks

Sketch the graphs of the following function:
$\text{f(x)}=2\sin\pi\text{x},0\leq\text{x}\leq2$

Answer

$\text{f(x)}=2\sin\pi\text{x},0\leq\text{x}\leq2$

$\text{x}$	$0$	$1$
$\text{f(x)}=2\sin\pi\text{x}$	$0$	$0$

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Question 43 Marks

Sketch the graphs of the following trigonometric functions:
$\text{g(x)}=\cos\Big(\text{x}+\frac{\pi}{4}\Big)$

Answer

$\text{g(x)}=\cos\Big(\text{x}+\frac{\pi}{4}\Big)$
$\text{y}=\cos\Big(\text{x}+\frac{\pi}{4}\Big)$
$\Rightarrow\text{y}-0=\cos\Big(\text{x}+\frac{\pi}{4}\Big) ...(\text{i})$
On shifting the origin at $\Big(-\frac{\pi}{4},0\Big),$ we get:
$\text{x}=\text{X}-\frac{\pi}{4}$ and $\text{y = Y}+0$
On subsitituting the values in (i), we get:
$\text{Y}=\cos\text{X}$
Then, we draw the graph of $\text{Y}=\cos\text{X}$ and shift it by $\frac{\pi}{4}$ to the left.
Then, we obtain the following graph:

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Question 53 Marks

Sketch the graphs of the following function:
$\text{h(x)}=2\sin3\text{x},0\leq\text{x}\leq\frac{2\pi}{3}$

Answer

$\text{h(x)}=2\sin3\text{x},0\leq\text{x}\leq\frac{2\pi}{3}$

$\text{x}$	$0$	$\frac{\pi}{3}$	$\frac{2\pi}{3}$
$\text{h(x)}=2\sin3\text{x}$	$0$	$0$	$0$

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Question 63 Marks

Sketch the graphs of the following curves on the same scale and the same axes:
$\text{y}=\cos\text{x}$ and $\text{y}=\cos\Big(\text{x}-\frac{\pi}{4}\Big)$

Answer

First, we draw the graph of $\text{y}=\cos\text{x}.$

Let us now draw the graph of $\text{y}=\cos\Big(\text{x}-\frac{\pi}4{}\Big).$

$\text{y}=\cos\Big(\text{x}-\frac{\pi}{4}\Big)$

$\Rightarrow\text{y}-0=\cos\Big(\text{x}-\frac{\pi}{4}\Big)...(\text{i})$

On shifting the origin at $\Big(\frac{\pi}{4},0\Big),$ we get:

$\text{x = X}+\frac{\pi}{4}$ and $\text{y = Y}+0$

On subsitituting the values in (i), we get:

$\text{Y}=\cos\text{X}$

Then, we draw the graph of $\text{Y}=\cos\text{X}$ and shift it by $\frac{\pi}{4}$ to the right.

Then, we will obtain the following graph: