What is the general solution for -sqrt(2)*sin(x)=sin(x)*cos(x) ?

The general solution for -sqrt(2)*sin(x)=sin(x)*cos(x) is x=2pin,x=pi+2pin

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-sqrt(2)sin(x)=sin(x)cos(x)

- 2 \cdot sin (x) = sin (x) \cdot cos (x)

x = 2 πn, x = π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

- 2 sin (x) = sin (x) cos (x)

Subtract

sin (x) cos (x)

from both sides

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

Factor

- 2 sin (x) - sin (x) cos (x) : - sin (x) (2 + cos (x))

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

Factor out common term

sin (x)

= - sin (x) (2 + cos (x))

- sin (x) (2 + cos (x)) = 0

Solving each part separately

sin (x) = 0 or 2 + cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

2 + cos (x) = 0 :

No Solution

2 + cos (x) = 0

Move

2

to the right side

2 + cos (x) = 0

Subtract

2

from both sides

2 + cos (x) - 2 = 0 - 2

Simplify

cos (x) = - 2

cos (x) = - 2

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

tan^{2} (x) + 3 sec (x) + 3 = 0, 0^{\circ} \leq x \leq 36 0^{\circ}

sin^{2} (α) = 1 + cos^{2} (α)

x, cos (2 y - 5 x) = 4 y + 5

21 cos (\frac{2 π}{687} t) + 228 = 220

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

What is the general solution for -sqrt(2)*sin(x)=sin(x)*cos(x) ?
The general solution for -sqrt(2)*sin(x)=sin(x)*cos(x) is x=2pin,x=pi+2pin