How do you use the half angle formula to solve for sin(x/2) when given sin(x)=(24/25)?

2 Answers
Jun 17, 2018

sin(x/2)=3/5 or sqrt(17)/5

Explanation:

angle x can be in first quadrant or second quadrant
1. first quadrant (0 < x < pi/2 )
cos x = 1 - sin^2(x) = sqrt(25^2-24^2)/25=7/25
2. second quadrant (pi/2 < x < pi )
cos x = -7/25

cos x = 1-2*sin^2(x/2)
sin(x/2)=sqrt((1-cos (x))/2)
(angle x/2 is in first quadrant. sin(x/2)>0)
sin (x/2)=sqrt(1/2((25+-7)/25)
sin(x/2)=3/5 or 4/5

Jun 17, 2018

sin(x/2)=3/5, or, 4/5.

Explanation:

We know that, cosx=1-2sin^2(x/2)..................(star).

But, sinx=24/25.

:. cos^2x=1-sin^2x=1-(24/25)^2=(25^2-24^2)/25^2.

:. cos^2x={(25+24)(25-24)}/25^2=49/25^2.

:. cosx=+-7/25.

Utilising this value of cosx in (star), we get,

2sin^2(x/2)=1-(+-7/25), i.e.,

2sin^2(x/2)=18/25, or, 2sin^2(x/2)=32/25.

:. sin^2(x/2)=9/25, or, 16/25.

Let us note that, sinx gt 0 :. 0 lt x lt pi :. 0 lt x/2 lt pi/2.

:. sin(x/2)" must be "+ve.

:. sin(x/2)=+3/5, or, +4/5.

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