Question

Sketch the graphs of the following trigonometric functions:

(i) u (x) = cos² x/2

(ii) f (x) = cos πx

(iii) g (x) = cos 2π x

Answer 1

(i) u (x) = cos² x/2

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

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

x	0 (A)	π (B)	2π (C)	3π (D)
u (x) = cos2 x/2	1	0	1	0

Thus, the required curve is

(ii) f (x) = cos πx

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

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

x	0 (A)	1/2 (B)	1 (C)	3/2 (D)	2E	5/2 (F)
f (x) = cos πx	1	0	-1	0	1	0

Thus, the required curve is

(iii) g(x) = cos 2π x

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

Therefore, g (x) = cos (2πx) is a periodic function with period 1. Therefore, we will draw the graph of g (x) = cos (2πx) in the interval [0, 1]. The values of g (x) = cos (2πx) at various points in [0, 1] are listed in the following table:

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

Thus, the required curve is