How to show \(\displaystyle{\tan{{\frac{{\pi}}{{{7}}}}}}{ < }{\sin{{\frac{{{5}\pi}}{{{18}}}}}}\) without

Lilliana Villa

Lilliana Villa

Answered question

2022-04-07

How to show tanπ7<sin5π18 without a calculator?
Attempts:Since both sides are greater then zero, we can compare the squares of the functions.Replaced tanπ7 with cot(π2π7)=cot5π14. Then I replaced (sin5π18)2 with (11+cot5π18)2. Then I tried to simplify the difference, but it didn't clarify anyhing. Could you please help me compare those functions?

Answer & Explanation

fonne0kgq

fonne0kgq

Beginner2022-04-08Added 11 answers

Note that sin(5π18)>sin(π4)=12 ,and tan(π7)<tan(π6)=13
Rodach42hj

Rodach42hj

Beginner2022-04-09Added 4 answers

It is not difficult since they are very distant from each other. The given inequality is equivalent to
sinπ7<sin5π18,cosπ7
or to
2sinπ7<sin53π126+sin17π126
which can be proved through
2sinπ7<2π7<109<sin53π126+sin17π126
since 18π<70  and  2πx<sinx<x  for  x(0,π2)

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