Use the identities cos &#x2061;<!-- ⁡ --> 2 x = 2 cos 2 </msup> &#x2061

Aditya Erickson 2022-05-23 Answered
Use the identities
cos 2 x = 2 cos 2 x 1 = 1 2 sin 2 x
sin x = cos ( π 2 x )
to help evaluate
0 π / 2 1 sin x d x
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Answers (2)

odczepneyv
Answered 2022-05-24 Author has 10 answers
So, if sin x = cos ( π 2 x ), then
x = 0 π / 2 1 sin x d x = x = 0 π / 2 1 cos ( π 2 x ) d x = u = 0 π / 2 1 cos u d u ,
the last equality due to the substitution u = π / 2 x. Now use the same method of solution, namely 1 cos u = 2 sin u 2 , to deduce the result.
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Zeihergp
Answered 2022-05-25 Author has 6 answers
0 π / 2 1 sin x d x
= 2 0 π / 2 1 sin x d ( x / 2 )
= 2 0 π / 2 ( sin x / 2 cos x / 2 ) 2 d ( x / 2 )
= 2 0 π / 2 ( sin x / 2 cos x / 2 ) d ( x / 2 )
that you can further on process taking x / 2 = u for substitution.
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