What is the general solution for sin(3x)=sqrt(3)cos(3x) ?

The general solution for sin(3x)=sqrt(3)cos(3x) is x= pi/9+(pin)/3

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sin(3x)=sqrt(3)cos(3x)

sin (3 x) = 3 cos (3 x)

x = \frac{π}{9} + \frac{πn}{3}

Degrees

x = 2 0^{\circ} + 6 0^{\circ} n

Solution steps

sin (3 x) = 3 cos (3 x)

Rewrite using trig identities

sin (3 x) = 3 cos (3 x)

Divide both sides by

cos (3 x), cos (3 x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

tan (3 x) = 3

tan (3 x) = 3

General solutions for

tan (3 x) = 3

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

3 x = \frac{π}{3} + πn

3 x = \frac{π}{3} + πn

Solve

3 x = \frac{π}{3} + πn : x = \frac{π}{9} + \frac{πn}{3}

3 x = \frac{π}{3} + πn

Divide both sides by

3

3 x = \frac{π}{3} + πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{π}{3}}{3} + \frac{πn}{3} : \frac{π}{9} + \frac{πn}{3}

\frac{\frac{π}{3}}{3} + \frac{πn}{3}

\frac{\frac{π}{3}}{3} = \frac{π}{9}

\frac{\frac{π}{3}}{3}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{π}{9}

= \frac{π}{9} + \frac{πn}{3}

x = \frac{π}{9} + \frac{πn}{3}

x = \frac{π}{9} + \frac{πn}{3}

x = \frac{π}{9} + \frac{πn}{3}

x = \frac{π}{9} + \frac{πn}{3}

(9 sin (2 x) + 9 cos (2 x))^{2} = 81

- 2 cos^{2} (θ) + sin (θ) - 3 = - 4

sec^{2} (x) - 9 sec (x) = 0

- 3 sin (2 θ) = 5 cos (2 θ)

(tan^{2} (θ) - 4) (2 cos (θ) + 1) = 0

What is the general solution for sin(3x)=sqrt(3)cos(3x) ?
The general solution for sin(3x)=sqrt(3)cos(3x) is x= pi/9+(pin)/3