Find complete set of real values of values of " a'' such that point P ( a, sin a) lies inside the triangle formed by lines
x-2y+2=0
x+y=0 and
x-y-PI=0

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{have}: \\ \mathrm{x}-2\mathrm{y}+2=0 ;\left(\mathrm{i}\right) \\ \mathrm{x}+\mathrm{y}=0 ;\left(\mathrm{ii}\right) \\ \mathrm{x}-\mathrm{y}=\mathrm{\pi } ;\left(\mathrm{iii}\right) \\ \mathrm{We} \mathrm{will} \mathrm{need} \mathrm{to} \mathrm{sketch} \mathrm{these} \mathrm{lines} \mathrm{to} \mathrm{make} \mathrm{question} \mathrm{easy}. \end{gathered}$$


$$\begin{gathered} \mathrm{If} \mathrm{two} \mathrm{points} \mathrm{lie} \mathrm{on} \mathrm{same} \mathrm{of} \mathrm{the} \mathrm{line}, \mathrm{then} \mathrm{both} \mathrm{points} \mathrm{will} \mathrm{given} \mathrm{same} \mathrm{sign} \\ \mathrm{when} \mathrm{they} \mathrm{are} \mathrm{put} \mathrm{in} \mathrm{the} \mathrm{line}. \mathrm{Hence} \mathrm{we} \mathrm{will} \mathrm{need} \mathrm{a} \mathrm{reference} \mathrm{point}. \\ \mathrm{Now} \mathrm{from} \mathrm{graph} \mathrm{you} \mathrm{can} \mathrm{cleatly} \mathrm{see} \mathrm{that} \left(1,1\right) \mathrm{lies} \mathrm{inside} \mathrm{the} \mathrm{triangle} \mathrm{hence} \\ \mathrm{we} \mathrm{will} \mathrm{use} \mathrm{it} \mathrm{as} \mathrm{reference} \mathrm{point}. \\ {\mathrm{L}}_1: \mathrm{x}-2\mathrm{y}+2=0 \\ {\mathrm{L}}_2: \mathrm{x}+\mathrm{y}=0 \\ {\mathrm{L}}_3: \mathrm{x}-\mathrm{y}-\mathrm{\pi }=0 \\ \mathrm{Now} \\ {\mathrm{L}}_1\left(1,1\right): 1-2+2=1>0 \\ \mathrm{Hence} \\ {\mathrm{L}}_1:\left(\mathrm{a},\mathrm{sina}\right)>0 \\ \Rightarrow \mathrm{a}-2\mathrm{sina}+2>0 \\ \Rightarrow \mathrm{a}>2\mathrm{sina}-2 \\ \mathrm{Maxium} \mathrm{value} \mathrm{of} \mathrm{sina} \mathrm{is} 1, \\ \Rightarrow \mathrm{a}>2\times 1-2 \\ \Rightarrow \mathrm{a}>2-2 \\ \Rightarrow \mathrm{a}>0 \\ {\mathrm{L}}_2\left(1,1\right): 1+1=2>0 \\ \Rightarrow {\mathrm{L}}_2:\left(\mathrm{a},\mathrm{sina}\right)>0 \\ \mathrm{a}+\mathrm{sina}>0 \\ \mathrm{a}>-\mathrm{sina} \\ \mathrm{Due} \mathrm{to} \mathrm{negative} \mathrm{sign}, \mathrm{R}.\mathrm{H}.\mathrm{S}. \mathrm{will} \mathrm{be} \mathrm{maximum}, \mathrm{when} \mathrm{sina} \mathrm{is} \mathrm{minimum} \mathrm{and} \mathrm{minimum} \mathrm{value} \mathrm{of} \mathrm{sina} \mathrm{is} -1. \\ \Rightarrow \mathrm{a}>-\left(-1\right) \\ \Rightarrow \mathrm{a}>1 \\ {\mathrm{L}}_3\left(1,1\right):1-1-\mathrm{\pi }=-\mathrm{\pi }<0 \\ \Rightarrow \mathrm{a}-\mathrm{sina}-\mathrm{\pi }<0 \\ \Rightarrow \mathrm{a}<\mathrm{sina}+\mathrm{\pi } \\ \mathrm{a} \mathrm{should} \mathrm{be} \mathrm{less} \mathrm{than} \mathrm{minimum} \mathrm{value} \mathrm{of} \mathrm{R}.\mathrm{H}.\mathrm{S}. \\ \Rightarrow \mathrm{a}<-1+\mathrm{\pi } \mathrm{o} \\ \mathrm{a}<\mathrm{\pi }-1 \\ \mathrm{So} \mathrm{we} \mathrm{have}: \\ \mathrm{a}>0 \mathrm{and} \mathrm{a}>1 \mathrm{and} \mathrm{a}<\mathrm{\pi }-1 \\ \mathrm{Taking} \mathrm{common} \mathrm{values} \mathrm{we} \mathrm{get}: \\ \mathbf{a}\mathbf{\in }\left(\mathbf{1}\mathbf{,}\mathbf{ }\mathbf{\pi }\mathbf{-}\mathbf{1}\right) \end{gathered}$$
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