Trigonometry
posted by Holly on .
Find all solutions of 4(cos(x))**24 sin(x)5=0 in the interval (6pi, 8pi)
I tried to work it out and got: 4(cos**2x)4cosx 9 = 0, but I can't figure out what cosx = from there to finish the problem.

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Remark :
( cos x ) ^ 2 = 1  ( sin x ) ^ 2
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4 ( cos x ) ^ 2  4 sin( x )  5 = 0
4 [ 1  ( sin x ) ^ 2 ]  4 sin( x )  5 = 0
4  4 ( sin x ) ^ 2  4 sin ( x )  5 = 0
 4 ( sin x ) ^ 2  4 sin ( x )  1 = 0 Multiply both sides by  1
4 ( sin x ) ^ 2 + 4 sin ( x ) + 1 = 0
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Remark :
( a + b ) ^ 2 = a ^ 2 + 2 a b + b ^ 2
So :
4 sin ( x ) ^ 2 + 4 sin ( x ) + 1 = [ 2 sin ( x ) + 1 ] ^ 2
Becouse :
[ 2 sin ( x ) + 1 ] ^ 2 =
[ 2 sin ( x ) ] ^ 2 + 2 * 2 sin ( x ) + 1 ^ 2 =
4 sin ( x ) ^ 2 + 4 sin ( x ) + 1
4 ( sin x ) ^ 2 + 4 sin ( x ) + 1 = 0 is same :
[ 2 sin ( x ) + 1 ] ^ 2 = 0
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[ 2 sin ( x ) + 1 ] ^ 2 = 0 Take the square root of both sides
2 sin ( x ) + 1 = 0 Subtract 1 from both sides
2 sin ( x ) =  1 Divide bboth sides by 2
sin ( x ) =  1 / 2
sin ( x ) =  1 / 2 for :
x = 7 pi / 6 = 210 °
and
x = 11 pi / 6 = 330 °
The period of sin ( x ) is 2 pi
So :
sin ( x ) =  1 / 2 for :
x = 2 n pi +7 pi / 6
and
x = 2 n pi + 11 pi / 6
n is an integer
In the interval ( 6 pi, 8 pi )
x = 2 * 3 pi + 7 pi / 6
and
x = 2 * 3 pi + 11 pi / 6
Solutions :
x = 6 pi + 7 pi / 6
and
x = 6 pi + 11 pi / 6
OR
x = 6 * 6 pi / 6 + 7 pi / 6 = 36 pi / 6 + 7 pi / 6 = 43 pi / 6
and
x = 6 * 6 pi / 6 + 11 pi / 6 = 36 pi / 6 + 11 pi / 6 = 47 pi / 6 
You apparently think that sinx = 1+cosx
The correct identity is cos^2 + sin^2 = 1
so, cos^2 x = 1sin^2 x
4(1sin^2 x)  4sinx  5 = 0
4  4sin^2 x  4sinx  5 = 0
4sin^2 x + 4sinx + 1 = 0
(2sinx + 1)^2 = 0
2sinx + 1 = 0
sinx = 1/2
Take it from there.