**Question:**

Sketch the graphs of the following pairs of functions on the same axes:

(i) $f(x)=\sin x, g(x)=\sin \left(x+\frac{\pi}{4}\right)$

(ii) $f(x)=\sin x, g(x)=\sin 2 x$

(iii) $f(x)=\sin 2 x, g(x)=2 \sin x$

(iv) $f(x)=\sin \frac{x}{2}, g(x)=\sin x$

**Solution:**

(i) $f(x)=\sin x, g(x)=\sin \left(x+\frac{\pi}{4}\right)$

Clearly, $\sin x$ and $\sin \left(x+\frac{\pi}{4}\right)$ is a periodic function with period $2 \pi$.

The graphs of $f(x)=\sin x$ and $g(x)=\sin \left(x+\frac{\pi}{4}\right)$ on different axes are shown below:

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

(ii) *f*(*x*) = sin* **x, g*(*x*) = sin 2*x*

Clearly, sin *x* and sin 2*x* is a periodic function with period 2π and π, respectively.

The graphs of *f*(*x*) = sin* **x *and* g*(*x*) = sin 2*x* on different axes are shown below:

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

(iii) *f*(*x*) = sin 2*x*, *g*(*x*) = 2 sin *x*

Clearly, sin 2*x* and 2 sin *x* is a periodic function with period π and 2π, respectively.

The graphs of *f*(*x*) = sin 2*x* and *g*(*x*) = 2 sin *x* on different axes are shown below:

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

(iv) $f(x)=\sin \frac{x}{2}, g(x)=\sin x$

Clearly, $\sin \frac{x}{2}$ and $\sin x$ is a periodic function with period $4 \pi$ and $2 \pi$, respectively.

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

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

Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.