What is the general solution for sin(x)(1+cos(x))=sin(x)+cos(x) ?

The general solution for sin(x)(1+cos(x))=sin(x)+cos(x) is x= pi/2+2pin,x=(3pi)/2+2pin

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sin(x)(1+cos(x))=sin(x)+cos(x)

sin (x) (1 + cos (x)) = sin (x) + cos (x)

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

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (x) (1 + cos (x)) = sin (x) + cos (x)

Subtract

sin (x) + cos (x)

from both sides

sin (x) cos (x) - cos (x) = 0

Factor

sin (x) cos (x) - cos (x) : cos (x) (sin (x) - 1)

sin (x) cos (x) - cos (x)

Factor out common term

cos (x)

= cos (x) (sin (x) - 1)

cos (x) (sin (x) - 1) = 0

Solving each part separately

cos (x) = 0 or sin (x) - 1 = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

General solutions for

cos (x) = 0

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

sin (x) - 1 = 0 : x = \frac{π}{2} + 2 πn

sin (x) - 1 = 0

Move

1

to the right side

sin (x) - 1 = 0

Add

1

to both sides

sin (x) - 1 + 1 = 0 + 1

Simplify

sin (x) = 1

sin (x) = 1

General solutions for

sin (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}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Combine all the solutions

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

9 0^{2} = 6 0^{2} + 7 0^{2} - 2 (60) (70) (cos (x))

sin^{2} (x) + 3 sin (x) + 7 = 5

cos (2 x) = - 0.6, - 18 0^{\circ} \leq x \leq 18 0^{\circ}

23 cos (\frac{θ}{2}) = - 3

\frac{1}{cos ^{2} ( \frac{θ}{2} )} = 12 cos (θ)

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