Solve the equation for Theta, (0 ≤ Theta < 2pi).?

Question

Solve the equation for Theta, (0 ≤ Theta < 2pi).

cos theta/2 - cos theta = 1

Answer 1

The equation is  cos θ/2  -  cos θ  =  1

To make for easier typing,  I will use  x for  ÃŽÂ¸/2, and 2x for θ.

So, cos x  - cos 2x  =  1

Using the compound angle formulae, substitute  (2cos²x - 1)  for  cos2x,

giving  cos x - (2 cos²x - 1)  =  1

Re-arranging, this becomes  2 cos²x  -  cos x  =  0

or  cos x(2 cos x  -  1)  =  0

Hence, either  cos x  =  0  or  (2cos x  -  1)  =  0

So  cos x  =  0  or cos x  =  Ã‚½

If cos x = 0 then  x  =  ÃÂ€/2  or x  =  3 π/2;  if cos x  =  Ã‚½  then x =   π/3  or  5 π/3

Since  x = θ/2, then  ÃŽÂ¸/2  =   π/2 ; 3 π/2 ;  ÃÂ€/3 ; or  5 π/3

Hence θ   =  ÃÂ€ ; 3π ;  2π/3 ;  or  10π/3 (which is the same as π/3)

Answer 2

cos t = cos^t/2 -sin^2t/2=2 cos^2t/2-1

so

cost/2-2cos^2t/2+1=1 so cost/2(1-2cost/2)=0

cost/2=0 t/2=pi/2+kpi

t= pi+2kpi for k=0

1-2cost/2=0  cos t/2=1/2  t/2=pi/3 +2kpi   t= 2pi/3+4kpi for k=0

and t/2=5pi/3+2kpi t=10pi/3+4kpi (no solution in the interval)