SOLUTION: find the exact solutions of the given equation in the interval [0 ,2TT] cos x/2- sin x = 0

Question 548058: find the exact solutions of the given equation in the interval [0 ,2TT] cos x/2- sin x = 0
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
find the exact solutions of the given equation in the interval [0 ,2TT] cos x/2- sin x = 0
**
cos x/2- sin x = 0
using cos half-angle formula
cos(x/2)=√[(1+cosx)/2]
√[(1+cosx)/2]-sinx=0
√[(1+cosx)/2]=sinx
square both sids
(1+cosx)/2=sin^x=1-cos^2x
1+cosx=2-2cos^2x
2cos^2x+cosx-1=0
(2cosx-1)(cosx+1)=0
..
(2cosx-1)=0
2cosx=1
cosx=1/2
x=π/3 and 5π/3 (in quadrants I and IV where cos>0
or
cosx+1=0
cosx=-1
x=π
ans:
Exact solutions:
x=π/3, 5π/3 and π
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