Solve trig equations using sum and difference identities

Question

1. find all solutions of sin(x + pi/2) + sin(x - 3pi/2)=1 in the interval [0,2pi)
2. find all solutions of sin(x + pi/3) + sin(x - pi/3)=1 in the interval [0,2pi)
3. find all solutions of cos(x + pi/6) + sin(x - pi/6)=1 in the interval [0,2pi)

Answer 1

Here is how the first one is done. THe second is done the eact same way...

sin (x + pi/2) = sin x cos (pi/2) + cos x sin(pi/2)

= sin x * 0 + cos x * 1

= cos x

sin (x - 3 pi /2) = sin x cos (3 pi/2) - cos x sin (3 pi /2 )

= sin x * 0 - cos x * -1

= cos x

so the equation becomes 2 * cos x = 1

cos x = 1/2

x = 60, 120, 240, 300

The third one, be careful, you need the angle addition formula for cosine:

cos(x + pi/6) = cos x cos (pi/6) - sin x sin (pi/6)

= cos x * sqrt(3)/2 - sin x * 1/2