Composition of Inverse Trigonometric Functions and Trigonometric Functions
Which of the ...
Question
Which of the following option is always correct for x∈(0,π2)?
A
sin(110)(110)<sin(19)(19)
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B
sin(110)(19)>sin(19)(110)
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C
sin(110)(110)>sin(19)(19)
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D
sin(110)(19)=sin(19)(110)
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Solution
The correct option is Csin(110)(110)>sin(19)(19) Let f(x)=sinxx
⇒f′(x)=xcosx−sinxx2
f′(x)=(cosxx2)(x−tanx)
⇒f′(x)<0∀x∈(0,π2)
(∵x<tanx in(0,π2)).
Hence f(x) is strictly decreasing
We know, (110)<(19)
⇒f(110)>f(19) ⇒sin(110)(110)>sin(19)(19)
Let f(x)=xsinx ⇒f′(x)=xcosx+sinx=cosx(x+tanx) ∵cosx,tanx>0 in x∈(0,π2) ⇒f′(x)>0,x∈(0,π2) ⇒f(x) strictly increases ⇒ for π2>x>0,f(x)>f(0) ⇒19sin(19)>110sin(110) ⇒sin(110)(19)<sin(19)(110)