11类RD Sharma解决方案–第六章三角函数图–练习6.2

问题1：绘制以下图形：

(i)y = cos(x + $\frac{\pi}{4}$ )

解决方案：

To obtain this graph y-0 = cos (x+$\frac{\pi}{4}$),

Shifting the origin at $(-\frac{\pi}{4},0)$, we have

X = x+$\frac{\pi}{4}$ and Y = y-0

Substituting these values, we get

Y = cos X

First we draw the graph of Y = cos X and shift it by $\frac{\pi}{4}$ to the left.

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

解决方案：

To obtain this graph y-0 = cos (x-$\frac{\pi}{4}$),

Shifting the origin at $(\frac{\pi}{4},0)$, we have

X = $x-\frac{\pi}{4}$ and Y = y-0

Substituting these values, we get

Y = cos X

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

(iii)y = $3 \hspace{0.1cm}cos (2x-1)$

解决方案：

To obtain this graph y-0 = 3 cos 2(x-$\frac{1}{2}$),

Shifting the origin at $(\frac{1}{2},0)$, we have

X = $x-\frac{1}{2}$ and Y = y-0

Substituting these values, we get

Y = 3 cos 2X

First we draw the graph of Y = 3 cos 2X and shift it by $\frac{1}{2}$ to the right.

The maximum and minimum values of y are 3 and -3 respectively.

(iv) $y = 2 \hspace{0.1cm}cos (x-\frac{\pi}{2})$

解决方案：

To obtain this graph y-0 = $2 \hspace{0.1cm}cos (x-\frac{\pi}{2})$,

Shifting the origin at $(\frac{\pi}{2},0)$, we have

X = $x-\frac{\pi}{2}$ and Y = y-0

Substituting these values, we get

Y = 2 cos X

First we draw the graph of Y = 2 cos X and shift it by $\frac{\pi}{2}$ to the right.

The maximum and minimum values of y are 2 and -2 respectively.

问题2：以相同比例绘制以下函数的图形：

(i)y = cos x，y = cos $(x-\frac{\pi}{4})$

解决方案：

Graph 1:

y = cos x

Graph 2:

To obtain this graph y-0 = cos $(x-\frac{\pi}{4})$,

Shifting the origin at $(\frac{\pi}{4},0)$, we have

X = $x-\frac{\pi}{4}$ and Y = y-0

Substituting these values, we get

Y = cos X

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

The graph y = cos x and $y = cos (x-\frac{\pi}{4})$ are on same axes are as follows:

(ii)y = cos 2x，y = cos $2(x-\frac{\pi}{4})$

解决方案：

Graph 1:

To obtain this graph y = cos 2x,

First we draw the graph of y =  cos x and then divide the x-coordinates of the points where it crosses x-axis by 2.

Graph 2:

To obtain this graph y-0 = cos $2(x-\frac{\pi}{4})$,

Shifting the origin at $(\frac{\pi}{4},0)$, we have

X = $x-\frac{\pi}{4}$ and Y = y-0

Substituting these values, we get

Y = cos 2X

First we draw the graph of Y = cos 2X and shift it by $\frac{\pi}{4}$ to the right.

The graph y = cos 2x and y = cos $2(x-\frac{\pi}{4})$are on same axes are as follows:

(iii)y = cos x， $y = cos (\frac{x}{2})$

解决方案：

Graph 1:

y = cos x

Graph 2:

To obtain this graph y = $cos (\frac{x}{2})$,

First we draw the graph of y =  cos x and then multiply the x-coordinates of the points where it crosses x-axis by 2.

The graph y = cos x and y = $cos (\frac{x}{2})$ are on same axes are as follows: