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

The general solution for sin(3x)=-2cos(3x) is x=-(1.10714…)/3+(pin)/3

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

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

x = - \frac{1.10714 \dots}{3} + \frac{πn}{3}

Degrees

x = - 21.14498 \dots^{\circ} + 6 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

tan (3 x) = - 2

tan (3 x) = - 2

Apply trig inverse properties

tan (3 x) = - 2

General solutions for

tan (3 x) = - 2

tan (x) = - a \Rightarrow x = arctan (- a) + πn

3 x = arctan (- 2) + πn

3 x = arctan (- 2) + πn

Solve

3 x = arctan (- 2) + πn : x = - \frac{arctan ( 2 )}{3} + \frac{πn}{3}

3 x = arctan (- 2) + πn

Simplify

arctan (- 2) + πn : - arctan (2) + πn

arctan (- 2) + πn

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 2) = - arctan (2)

= - arctan (2) + πn

3 x = - arctan (2) + πn

Divide both sides by

3

3 x = - arctan (2) + πn

Divide both sides by

3

\frac{3 x}{3} = - \frac{arctan ( 2 )}{3} + \frac{πn}{3}

Simplify

x = - \frac{arctan ( 2 )}{3} + \frac{πn}{3}

x = - \frac{arctan ( 2 )}{3} + \frac{πn}{3}

x = - \frac{arctan ( 2 )}{3} + \frac{πn}{3}

Show solutions in decimal form

x = - \frac{1.10714 \dots}{3} + \frac{πn}{3}

2 cos (3 x) = 1 + cos (x)

2 csc (x) - 2 = 0

90 = sin (x)

sin (2 x) csc (x) = 1, 0 \leq x \leq 2 π

tan (b) = 3

What is the general solution for sin(3x)=-2cos(3x) ?
The general solution for sin(3x)=-2cos(3x) is x=-(1.10714…)/3+(pin)/3