What is the general solution for solvefor a,cos(ax)+(b/x)sin(ax)=0 ?

The general solution for solvefor a,cos(ax)+(b/x)sin(ax)=0 is a=(arctan(-x/b))/x+(pin)/x

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Popular >

solvefor a,cos(ax)+(b/x)sin(ax)=0

Solution

solve for

a, cos (a x) + (\frac{b}{x}) sin (a x) = 0

Solution

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

Solution steps

cos (a x) + (\frac{b}{x}) sin (a x) = 0

Rewrite using trig identities

cos (a x) + \frac{b}{x} sin (a x) = 0

Divide both sides by

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

\frac{cos ( a x ) + \frac{b}{x} sin ( a x )}{cos ( a x )} = \frac{0}{cos ( a x )}

Simplify

1 + \frac{b sin ( a x )}{x cos ( a x )} = 0

Use the basic trigonometric identity:

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

1 + \frac{b tan ( a x )}{x} = 0

1 + \frac{b tan ( a x )}{x} = 0

Move

1

to the right side

1 + \frac{b tan ( a x )}{x} = 0

Subtract

1

from both sides

1 + \frac{b tan ( a x )}{x} - 1 = 0 - 1

Simplify

\frac{b tan ( a x )}{x} = - 1

\frac{b tan ( a x )}{x} = - 1

Multiply both sides by

x

\frac{b tan ( a x )}{x} = - 1

Multiply both sides by

x

\frac{b tan ( a x ) x}{x} = (- 1) x; x \neq = 0

Simplify

b tan (a x) = - x; x \neq = 0

b tan (a x) = - x; x \neq = 0

Divide both sides by

b; x \neq = 0, b \neq = 0

b tan (a x) = - x; x \neq = 0

Divide both sides by

b; x \neq = 0, b \neq = 0

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

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

a x = arctan (- \frac{x}{b}) + πn

a x = arctan (- \frac{x}{b}) + πn

Solve

a x = arctan (- \frac{x}{b}) + πn : a = \frac{arctan ( - \frac{x}{b} )}{x} + \frac{πn}{x}; x \neq = 0

a x = arctan (- \frac{x}{b}) + πn

Divide both sides by

x; x \neq = 0

a x = arctan (- \frac{x}{b}) + πn

Divide both sides by

x; x \neq = 0

\frac{a x}{x} = \frac{arctan ( - \frac{x}{b} )}{x} + \frac{πn}{x}; x \neq = 0

Simplify

a = \frac{arctan ( - \frac{x}{b} )}{x} + \frac{πn}{x}; x \neq = 0

a = \frac{arctan ( - \frac{x}{b} )}{x} + \frac{πn}{x}; x \neq = 0

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

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor a,cos(ax)+(b/x)sin(ax)=0 ?
The general solution for solvefor a,cos(ax)+(b/x)sin(ax)=0 is a=(arctan(-x/b))/x+(pin)/x