What is the general solution for sin(2x)=sqrt(3)cos(x),0<= x<= 2pi ?

Q: What is the general solution for sin(2x)=sqrt(3)cos(x),0<= x<= 2pi ?

The general solution for sin(2x)=sqrt(3)cos(x),0<= x<= 2pi is x= pi/2 ,x=(3pi)/2 ,x= pi/3 ,x=(2pi)/3

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sin(2x)=sqrt(3)cos(x),0<= x<= 2pi

Solution

sin (2 x) = 3 cos (x), 0 \leq x \leq 2 π

Solution

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

Degrees

x = 9 0^{\circ}, x = 27 0^{\circ}, x = 6 0^{\circ}, x = 12 0^{\circ}

Solution steps

sin (2 x) = 3 cos (x), 0 \leq x \leq 2 π

Subtract

3 cos (x)

from both sides

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

Rewrite using trig identities

sin (2 x) - cos (x) 3

Use the Double Angle identity:

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

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

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

Factor

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

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

Factor out common term

cos (x)

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

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

Solving each part separately

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

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

cos (x) = 0, 0 \leq x \leq 2 π

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

Solutions for the range

0 \leq x \leq 2 π

x = \frac{π}{2}, x = \frac{3 π}{2}

- 3 + 2 sin (x) = 0, 0 \leq x \leq 2 π : x = \frac{π}{3}, x = \frac{2 π}{3}

- 3 + 2 sin (x) = 0, 0 \leq x \leq 2 π

Move

3

to the right side

- 3 + 2 sin (x) = 0

Add

3

to both sides

- 3 + 2 sin (x) + 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}

General solutions for

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

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}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

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

Solutions for the range

0 \leq x \leq 2 π

x = \frac{π}{3}, x = \frac{2 π}{3}

Combine all the solutions

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

Popular Examples

2tan(A)-1=0

2 tan (A) - 1 = 0

cos^2(t)+9/49 =1

cos^{2} (t) + \frac{9}{49} = 1

4tan(x)+cot(x)=5

4 tan (x) + cot (x) = 5

75=arcsin((1.4)/x)

75 = arcsin (\frac{1.4}{x})

tan(2x+60)=3sqrt(3)

tan (2 x + 60) = 33

Frequently Asked Questions (FAQ)

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