What is the general solution for 2sin(x-pi/2)+1=0 ?

The general solution for 2sin(x-pi/2)+1=0 is x=2pin+(5pi)/3 ,x=2pin+(7pi)/3

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2sin(x-pi/2)+1=0

2 sin (x - \frac{π}{2}) + 1 = 0

x = 2 πn + \frac{5 π}{3}, x = 2 πn + \frac{7 π}{3}

Solution steps

2 sin (x - \frac{π}{2}) + 1 = 0

Move

1

to the right side

2 sin (x - \frac{π}{2}) = - 1

Divide both sides by

2

sin (x - \frac{π}{2}) = - \frac{1}{2}

General solutions for

sin (x - \frac{π}{2}) = - \frac{1}{2}

x - \frac{π}{2} = \frac{7 π}{6} + 2 πn, x - \frac{π}{2} = \frac{11 π}{6} + 2 πn

Solve

x - \frac{π}{2} = \frac{7 π}{6} + 2 πn : x = 2 πn + \frac{5 π}{3}

Solve

x - \frac{π}{2} = \frac{11 π}{6} + 2 πn : x = 2 πn + \frac{7 π}{3}

x = 2 πn + \frac{5 π}{3}, x = 2 πn + \frac{7 π}{3}

2 cos^{2} (x) - 1 = cos (x)

0 = 2 tan (x) - sec^{2} (x)

csc^{2} (x) + 6 csc (x) - 7 = 0

\frac{sin ( 2 x ) - 1}{3} = - \frac{1}{6}

2 sin^{2} (2 x) + cos (2 x) - 1 = 0

What is the general solution for 2sin(x-pi/2)+1=0 ?
The general solution for 2sin(x-pi/2)+1=0 is x=2pin+(5pi)/3 ,x=2pin+(7pi)/3