find all solutions in the interval for these equations

Question

find all solutions in the interval for these equations

1. [0,2pi] to cos x=√2/2

 

2. [0,2pi] to cos x sin x-cos x=0

 

3. [0,2pi] to 2sin² x+sin x= 1

Answer 1

1) cos x = square root of two, divided by 2, tells us that x = pi/4 or 7pi/4

 

2) cos x(sin x - cos x) = 0 occurs whenever cos x = 0, i.e. pi/2 & 3pi/2, and also when sin x = cos x which occurs at pi/4 and 5pi/4.

 

3) 2 sin²x + sin x - 1 = 0; factor  (2sin x -1)(sin x +1) = 0.  This is true when sin x = -1, i.e. 3pi/2 and when sin x = ½ which gives x = pi/6 or 5pi/6.