What is the general solution for sin((3pi)/2+x)+sin((3pi)/2+x)=-2 ?

The general solution for sin((3pi)/2+x)+sin((3pi)/2+x)=-2 is x=2pin

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

sin (\frac{3 π}{2} + x) + sin (\frac{3 π}{2} + x) = - 2

x = 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (\frac{3 π}{2} + x) + sin (\frac{3 π}{2} + x) = - 2

Simplify

sin (\frac{3 π}{2} + x) + sin (\frac{3 π}{2} + x) : 2 sin (\frac{3 π}{2} + x)

sin (\frac{3 π}{2} + x) + sin (\frac{3 π}{2} + x)

Add similar elements:

sin (x + \frac{3 π}{2}) + sin (x + \frac{3 π}{2}) = 2 sin (x + \frac{3 π}{2})

= 2 sin (x + \frac{3 π}{2})

2 sin (\frac{3 π}{2} + x) = - 2

Divide both sides by

2

2 sin (\frac{3 π}{2} + x) = - 2

Divide both sides by

2

\frac{2 sin ( \frac{3 π}{2} + x )}{2} = \frac{- 2}{2}

Simplify

sin (\frac{3 π}{2} + x) = - 1

sin (\frac{3 π}{2} + x) = - 1

General solutions for

sin (\frac{3 π}{2} + x) = - 1

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{3 π}{2} + x = \frac{3 π}{2} + 2 πn

\frac{3 π}{2} + x = \frac{3 π}{2} + 2 πn

Solve

\frac{3 π}{2} + x = \frac{3 π}{2} + 2 πn : x = 2 πn

\frac{3 π}{2} + x = \frac{3 π}{2} + 2 πn

Subtract

\frac{3 π}{2}

from both sides

\frac{3 π}{2} + x - \frac{3 π}{2} = \frac{3 π}{2} + 2 πn - \frac{3 π}{2}

Simplify

x = 2 πn

x = 2 πn

cos (x) tan (x) = 0

cos^{2} (x) + 4 cos (x) + 3 = 0

sin (x) + sin^{2} (x) = 0

2 sin^{2} (x) - sin (x) - 1 = 0, 0 \leq x < 2 π

sin^{2} (x) = 3

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