Question

Sketch the graphs of the following trigonometric functions:

(i) f (x) = cos (x – π/4)

(ii) g (x) = cos (x + π/4)

(iii) h (x) = cos² 2x

Answer 1

(i) f (x) = cos (x – π/4)

As we know that g (x) = cos x is a periodic function with period 2π.

Therefore, f (x) = cos (x – π/4) is a periodic function with period π. Therefore, we will draw the graph of f (x) = cos (x – π/4) in the interval [0, π]. The values of f (x) = cos (x – π/4) at various points in [0, π] are listed in the following table:

x	0(A)	π/4 (B)	π/2 (C)	3π/4 (D)	π (E)	5π/4 (F)	3π/2 (G)	7π/4 (H)
f (x) = cos (x – π/4)	1/√2 = 0.7	1	1/√2 = 0.7	0	-1/√2 = -0.7	-1	-1/√2 = -0.7	0

Thus, the required curve is

(ii) g (x) = cos (x + π/4)

As we know that f (x) = cos x is a periodic function with period 2π.

Therefore, g (x) = cos (x + π/4) is a periodic function with period π. Therefore, we will draw the graph of g (x) = cos (x + π/4) in the interval [0, π]. The values of g (x) = cos (x + π/4) at various points in [0, π] are listed in the following table:

x	0 (A)	π/4 (B)	π/2 (C)	3π/4 (D)	π (E)	5π/4 (F)	3π/2 (G)	7π/4 (H)
g (x) = cos (x + π/4)	1/√2 = 0.7	0	-1/√2 = -0.7	-1	-1/√2 = -0.7	0	1/√2 = 0.7	1

Thus, the required curve is

(iii) h (x) = cos² 2x

As we know that f (x) = cos x is a periodic function with period 2π.

Therefore, h (x) = cos² 2x is a periodic function with period π. Therefore, we will draw the graph of h (x) = cos² 2x in the interval [0, π]. The values of h (x) = cos² 2x at various points in [0, π] are listed in the following table:

x	0 (A)	π/4 (B)	π/2 (C)	3π/4 (D)	π (E)	5π/4 (F)	3π/2 (G)
h (x) = cos2 2x	1	0	1	0	1	0	1

Thus, the required curve is