Trig double and half angle identities and solving equations

Question

I don't understand how to find x in the following :

 

2cos (x/2)-1=0

 

2sin 2x +1=0

 

 

Answer 1

2cos (x/2)-1=0.   cos (x/2) = 1/2.  We know that the angles having cosine equal to one half are ±60⁰ + k•360^o.

 

Therefore x/2 = the above, so if we double this relationship we get x = ±120⁰ + k•720^o.

 

There are infinitely many answers, depending on choice of integer k.

 

You can do the second problem by a similar approach.