Solve each equation on the interval 0 lessthanorequalto thet

Solve each equation on the interval 0 lessthanorequalto theta lessthanorequalto 2 pi sin(2 theta) - sin theta - 2cos theta + 1 = 0 cos(x + pi/4) - cos(x - pi/4) = 2 cos5x - cos3x = 0

Solution

I will answer a and b post c separately

a). I am taking theta as \'x\' , so i will be solving for value of x

sin(2x) -sinx -2cosx +1=0

but sin2x =2sinx.cosx

2sinx.cosx -sinx-2cosx+1=0

sinx(2cosx -1) -1(2cosx-1)=0

(2cosx-1) (sinx -1) =0

so (2cosx -1) =0 or (sinx -1) ==0

cosx =1/2 or sinx =1

x = pi/3 , 5pi/3 or x =pi/2

so x values are pi/3,pi/2,5pi/3

b).

cos(x+pi/4) -cos(x-pi/4) =0

cos(a+b) =cosa.cosb - sina.sinb

cos(x).cos(pi/4) - sin(x).sin(pi/4) - ( cosx.cospi/4 +sinx.sinpi/4) =0

so now we get -2sinx.sinpi/4=0

sinx =0

x = 0 or pi or 2pi