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

The general solution for cos(3x)=sqrt(3)sin(3x) is x= pi/(18)+(pin)/3

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Popular >

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

Solution

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

Solution

x = \frac{π}{18} + \frac{πn}{3}

Degrees

x = 1 0^{\circ} + 6 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

1 = 3 tan (3 x)

Switch sides

3 tan (3 x) = 1

3 tan (3 x) = 1

Divide both sides by

3

3 tan (3 x) = 1

Divide both sides by

3

\frac{3 tan ( 3 x )}{3} = \frac{1}{3}

Simplify

\frac{3 tan ( 3 x )}{3} = \frac{1}{3}

Simplify

\frac{3 tan ( 3 x )}{3} : tan (3 x)

\frac{3 tan ( 3 x )}{3}

Cancel the common factor:

3

= tan (3 x)

Simplify

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

tan (3 x) = \frac{3}{3}

tan (3 x) = \frac{3}{3}

tan (3 x) = \frac{3}{3}

General solutions for

tan (3 x) = \frac{3}{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}{6} + πn

3 x = \frac{π}{6} + πn

Solve

3 x = \frac{π}{6} + πn : x = \frac{π}{18} + \frac{πn}{3}

3 x = \frac{π}{6} + πn

Divide both sides by

3

3 x = \frac{π}{6} + πn

Divide both sides by

3

\frac{3 x}{3} = \frac{\frac{π}{6}}{3} + \frac{πn}{3}

Simplify

\frac{3 x}{3} = \frac{\frac{π}{6}}{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}{6}}{3} + \frac{πn}{3} : \frac{π}{18} + \frac{πn}{3}

\frac{\frac{π}{6}}{3} + \frac{πn}{3}

\frac{\frac{π}{6}}{3} = \frac{π}{18}

\frac{\frac{π}{6}}{3}

Apply the fraction rule:

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

= \frac{π}{6 \cdot 3}

Multiply the numbers:

6 \cdot 3 = 18

= \frac{π}{18}

= \frac{π}{18} + \frac{πn}{3}

x = \frac{π}{18} + \frac{πn}{3}

x = \frac{π}{18} + \frac{πn}{3}

x = \frac{π}{18} + \frac{πn}{3}

x = \frac{π}{18} + \frac{πn}{3}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos(3x)=sqrt(3)sin(3x) ?
The general solution for cos(3x)=sqrt(3)sin(3x) is x= pi/(18)+(pin)/3