What is the general solution for sin(2x)-3cos(x)=0 ?

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

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sin(2x)-3cos(x)=0

sin (2 x) - 3 cos (x) = 0

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 (2 x) - 3 cos (x) = 0

Rewrite using trig identities

sin (2 x) - 3 cos (x)

Use the Double Angle identity:

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

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

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

Factor

- 3 cos (x) + 2 cos (x) sin (x) : cos (x) (2 sin (x) - 3)

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

Factor out common term

cos (x)

= cos (x) (- 3 + 2 sin (x))

cos (x) (2 sin (x) - 3) = 0

Solving each part separately

cos (x) = 0 or 2 sin (x) - 3 = 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

2 sin (x) - 3 = 0 :

No Solution

2 sin (x) - 3 = 0

Move

3

to the right side

2 sin (x) - 3 = 0

Add

3

to both sides

2 sin (x) - 3 + 3 = 0 + 3

Simplify

2 sin (x) = 3

2 sin (x) = 3

Divide both sides by

2

2 sin (x) = 3

Divide both sides by

2

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

Simplify

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

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

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

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

5 cos (x) + 1 = 1

cosh (z) = 1

cos (2 x - \frac{π}{6}) = - \frac{1}{2}, \frac{π}{2} < x < π

sin (2 θ) + sin (θ) + 2 cos (θ) + 1 = 0

tan (θ) = \frac{563}{2000}

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