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Question

Find the general solutions of each of the following equations:
(i) cos 4x=cos 2x
(ii) cos 3x=sin 2x
(iii) sin mx+sin nx=0

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Solution

(i)
cos4x=cos2x

cos4xcos2x=0

2sin(4x+2x2)sin(4x2x2)

=2sin3xsinx=0

sin3x=0,sinx=0

General solution of sin3x

3x=nπ

x=nπ3

General solution of sinx

x=nπ



(ii)
cos3x=sin2x

cos3x=cos(π22x)

±3x=π22x

(a) π2=5x

x=π10

(b) π2=x

x=π2

cos3x=cos(π22x2π)

±3x=π22x2π

(c) π22π=5x

x=310π

(d) π22πx

x=32π



(iii)
sinmx+sinnx=0

2sin(mx+nx2)cos(mxnx2)=0

sin(mx+nx2)=0,cos(mxnx2)=0

(m+n2)x=px,(mn2)x=(2q+1)π2

x=2pxm+n,x=(2q+1)πmn

p and q are integers.


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