HCU MCA 2013 Paper

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Question : 39 of 75

Marks: +1, -0

For real numbers x, y and n ∈ Z , sin x = sin y implies which of the following ?

x = nπ + $(-1)^n$ y
x =2nπ ± y
x = nπ + y
x = nπ ± y

Solution:

(a) We have,
sin x = sin y ⇒ sin x - sin y = 0

\;\;\;\;\;\;\;2 \cos({\x+\y}/2)-\sin({\x-\y}/2)=0

⇒

\cos({\x+\y}/2)=0 \or\sin({\x-\y}/2)=0

⇒({\x+\y}/2)=(2\n+1)\π/2 \or ({\x-\y}/2)=\n\π

⇒({\x+\y})=(2\n+1)\π\;\;\;\;\or\;\;\; ({\x-\y})=2\n\π

⇒({\x})=[(2\n+1)\π-\y]\;\;\;\or\;\;\; ({\x})=(2\n\π+\y)

⇒

\;\;\;

x = [(an odd multiple of π) - y]
or

\;\;\;

x = [(an even multiple ofπ+ y]
⇒

\;\;\;

θ = nπ+ (-1)

^\n \y

, where n ∈ Z

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