What is the general solution for -13a*sin(2x)-13b*cos(2x)=0 ?

The general solution for -13a*sin(2x)-13b*cos(2x)=0 is x=(arctan(-b/a))/2+(pin)/2

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-13asin(2x)-13bcos(2x)=0

Solution

- 13 a \cdot sin (2 x) - 13 b \cdot cos (2 x) = 0

Solution

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

Solution steps

- 13 a sin (2 x) - 13 b cos (2 x) = 0

Rewrite using trig identities

- 13 a sin (2 x) - 13 b cos (2 x) = 0

Divide both sides by

cos (2 x), cos (2 x) \neq = 0

\frac{- 13 a sin ( 2 x ) - 13 b cos ( 2 x )}{cos ( 2 x )} = \frac{0}{cos ( 2 x )}

Simplify

- \frac{13 a sin ( 2 x )}{cos ( 2 x )} - 13 b = 0

Use the basic trigonometric identity:

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

- 13 a tan (2 x) - 13 b = 0

- 13 a tan (2 x) - 13 b = 0

Move

13 b

to the right side

- 13 a tan (2 x) - 13 b = 0

Add

13 b

to both sides

- 13 a tan (2 x) - 13 b + 13 b = 0 + 13 b

Simplify

- 13 a tan (2 x) = 13 b

- 13 a tan (2 x) = 13 b

Divide both sides by

- 13 a; a \neq = 0

- 13 a tan (2 x) = 13 b

Divide both sides by

- 13 a; a \neq = 0

\frac{- 13 a tan ( 2 x )}{- 13 a} = \frac{13 b}{- 13 a}; a \neq = 0

Simplify

\frac{- 13 a tan ( 2 x )}{- 13 a} = \frac{13 b}{- 13 a}

Simplify

\frac{- 13 a tan ( 2 x )}{- 13 a} : tan (2 x)

\frac{- 13 a tan ( 2 x )}{- 13 a}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{13 a tan ( 2 x )}{13 a}

Divide the numbers:

\frac{13}{13} = 1

= \frac{a tan ( 2 x )}{a}

Cancel the common factor:

a

= tan (2 x)

Simplify

\frac{13 b}{- 13 a} : - \frac{b}{a}

\frac{13 b}{- 13 a}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{13 b}{13 a}

Divide the numbers:

\frac{13}{13} = 1

= - \frac{b}{a}

tan (2 x) = - \frac{b}{a}; a \neq = 0

tan (2 x) = - \frac{b}{a}; a \neq = 0

tan (2 x) = - \frac{b}{a}; a \neq = 0

Apply trig inverse properties

tan (2 x) = - \frac{b}{a}

General solutions for

tan (2 x) = - \frac{b}{a}

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

2 x = arctan (- \frac{b}{a}) + πn

2 x = arctan (- \frac{b}{a}) + πn

Solve

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

2 x = arctan (- \frac{b}{a}) + πn

Divide both sides by

2

2 x = arctan (- \frac{b}{a}) + πn

Divide both sides by

2

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

Simplify

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

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

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

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Frequently Asked Questions (FAQ)

What is the general solution for -13a*sin(2x)-13b*cos(2x)=0 ?
The general solution for -13a*sin(2x)-13b*cos(2x)=0 is x=(arctan(-b/a))/2+(pin)/2