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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\left(\frac{x+y}{2}\right)-\sin\left(\frac{x-y}{2}\right)=0

⇒

\cos\left(\frac{x+y}{2}\right)=0\text{ or }\sin\left(\frac{x-y}{2}\right)=0

\Rightarrow \frac{x+y}{2}=\frac{(2n+1)\pi}{2}\text{ or }\frac{x-y}{2}=n\pi

\Rightarrow x+y=(2n+1)\pi\text{ or }x-y=2n\pi

\Rightarrow x=[(2n+1)\pi - y]\text{ or }x=(2n\pi + y)

⇒

\;\;\;

x = [(an odd multiple of π) - y]

or

\;\;\;

x = [(an even multiple ofπ+ y]

⇒

\;\;\;

θ = nπ+ (-1)

y^{n}

, where n ∈ Z

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