Question -
Answer -
(i) y = cos x and y = cos (x – π/4)
We know that the functions y = cos x and y = cos (x – π/4) are periodic functions with periods π and π.
The values of these functions are tabulated below:
| x | 0 | π/4 | π/2 | 3π/4 | π | 5π/4 | 3π/2 | 7π/4 |
| y = cos x | 1 | 1/√2 = 0.7 | 0 | -1/√2 = -0.7 | -1 | -1/√2 = -0.7 | 0 | 1 |
| y = cos (x – π/4) | 1/√2 = 0.7 | 1 | 1/√2 = 0.7 | 0 | -1/√2 = -0.7 | -1 | -1/√2 = -0.7 | 0 |
The required curve is:
(ii) y = cos 2x and y = cos 2(x – π/4)
We know that the functions y = cos 2x and y = cos 2(x – π/4) are periodic functions with periods π and π.
The values of these functions are tabulated below:
| x | 0 | π/4 | π/2 | 3π/4 | π | 5π/4 | 3π/2 | 7π/4 |
| y = cos x | 1 | 0 | -1 | 0 | 1 | 0 | -1 | 0 |
| y = cos 2 (x – π/4) | 0 | 1 | 0 | -1 | 0 | 1 | 0 | -1 |
The required curve is:
(iii) y = cos x and y = cos x/2
We know that the functions y = cos x and y = cos (x/2) are periodic functions with periods π and π.
The values of these functions are tabulated below:
| x | 0 | π/2 | π | 3π/2 | 2π |
| y = cos x | 1 | 0 | -1 | 0 | 1 |
| y = cos x/2 | 1 | 1/√2 = 0.7 | 0 | -1/√2 = -0.7 | -1 |
The required curve is:
(iv) y = cos2 x and y = cos x
We know that the functions y = cos2 xand y = cos x are periodic functions with period 2π.
The values of these functions are tabulated below:
| x | 0 | π/2 | π | 3π/2 | 2π |
| y = cos2 x | 1 | 0 | 1 | 0 | 1 |
| y = cos x | 1 | 0 | -1 | 0 | 1 |
The required curve is:
