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Bruno Pittman

Bruno Pittman

Answered question

2022-07-10

If
(1) sin y + cos y = 1 2
Then find
(2) x = sin 3 y cos 2 y + cos 3 y sin 2 y

Answer & Explanation

Bruno Dixon

Bruno Dixon

Beginner2022-07-11Added 14 answers

How about the following way?
Let s = sin y , c = cos y
Squaring the both sides of s + c = 1 / 2 gives
s 2 + 2 s c + c 2 = 1 4 s c = 3 8 .
Hence,
s 3 c 2 + c 3 s 2 = s 5 + c 5 ( s c ) 2 = ( s 2 + c 2 ) ( s 3 + c 3 ) s 2 c 2 ( s + c ) ( s c ) 2 = s 3 + c 3 ( s c ) 2 / 2 ( s c ) 2 = ( s + c ) ( s 2 s c + c 2 ) ( s c ) 2 / 2 ( s c ) 2 = ( 1 / 2 ) ( 1 s c ) ( s c ) 2 / 2 ( s c ) 2 = 79 18
Blericker74

Blericker74

Beginner2022-07-12Added 5 answers

Write
x = s ( s 2 / c 2 ) + c ( c 2 / s 2 )
then replace c 2 = 1 s 2 and vice versa to get
x = s ( 1 c 2 ) / c 2 + c ( 1 s 2 ) / s 2 = ( s / c 2 ) s + ( c / s 2 ) c
so that
x + s + c = s c 2 + c s 2 = s c 2 + c s 2 c 2 s 2 = c + s c s
Since c + s = 1 / 2, this gives
x + 1 2 = 1 2 c s
so
2 x + 1 = 1 c s
Now
( c + s ) = 1 / 2 ( c + s ) 2 = 1 / 4 c 2 + 2 c s + s 2 = 1 / 4 2 c s + 1 = 1 / 4 2 c s = 3 / 4 c s = 3 / 8
so the formula above becomes
2 x + 1 = 1 3 / 8 = 8 3
which you can solve.

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