What is the general solution for (4sin(x)-3cos(x))/(3sin(x)+4cos(x))=3 ?

The general solution for (4sin(x)-3cos(x))/(3sin(x)+4cos(x))=3 is x=-1.24904…+pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

(4sin(x)-3cos(x))/(3sin(x)+4cos(x))=3

Solution

\frac{4 sin ( x ) - 3 cos ( x )}{3 sin ( x ) + 4 cos ( x )} = 3

Solution

x = - 1.24904 \dots + πn

Degrees

x = - 71.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

\frac{4 sin ( x ) - 3 cos ( x )}{3 sin ( x ) + 4 cos ( x )} = 3

Subtract

3

from both sides

\frac{4 sin ( x ) - 3 cos ( x )}{3 sin ( x ) + 4 cos ( x )} - 3 = 0

Simplify

\frac{4 sin ( x ) - 3 cos ( x )}{3 sin ( x ) + 4 cos ( x )} - 3 : \frac{- 5 sin ( x ) - 15 cos ( x )}{3 sin ( x ) + 4 cos ( x )}

\frac{4 sin ( x ) - 3 cos ( x )}{3 sin ( x ) + 4 cos ( x )} - 3

Convert element to fraction:

3 = \frac{3 ( 3 sin ( x ) + 4 cos ( x ) )}{3 sin ( x ) + 4 cos ( x )}

= \frac{4 sin ( x ) - 3 cos ( x )}{3 sin ( x ) + 4 cos ( x )} - \frac{3 ( 3 sin ( x ) + 4 cos ( x ) )}{3 sin ( x ) + 4 cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{4 sin ( x ) - 3 cos ( x ) - 3 ( 3 sin ( x ) + 4 cos ( x ) )}{3 sin ( x ) + 4 cos ( x )}

Expand

4 sin (x) - 3 cos (x) - 3 (3 sin (x) + 4 cos (x)) : - 5 sin (x) - 15 cos (x)

4 sin (x) - 3 cos (x) - 3 (3 sin (x) + 4 cos (x))

Expand

- 3 (3 sin (x) + 4 cos (x)) : - 9 sin (x) - 12 cos (x)

- 3 (3 sin (x) + 4 cos (x))

Apply the distributive law:

a (b + c) = ab + a c

a = - 3, b = 3 sin (x), c = 4 cos (x)

= - 3 \cdot 3 sin (x) + (- 3) \cdot 4 cos (x)

Apply minus-plus rules

+ (- a) = - a

= - 3 \cdot 3 sin (x) - 3 \cdot 4 cos (x)

Simplify

- 3 \cdot 3 sin (x) - 3 \cdot 4 cos (x) : - 9 sin (x) - 12 cos (x)

- 3 \cdot 3 sin (x) - 3 \cdot 4 cos (x)

Multiply the numbers:

3 \cdot 3 = 9

= - 9 sin (x) - 3 \cdot 4 cos (x)

Multiply the numbers:

3 \cdot 4 = 12

= - 9 sin (x) - 12 cos (x)

= - 9 sin (x) - 12 cos (x)

= 4 sin (x) - 3 cos (x) - 9 sin (x) - 12 cos (x)

Simplify

4 sin (x) - 3 cos (x) - 9 sin (x) - 12 cos (x) : - 5 sin (x) - 15 cos (x)

4 sin (x) - 3 cos (x) - 9 sin (x) - 12 cos (x)

Add similar elements:

- 3 cos (x) - 12 cos (x) = - 15 cos (x)

= 4 sin (x) - 15 cos (x) - 9 sin (x)

Add similar elements:

4 sin (x) - 9 sin (x) = - 5 sin (x)

= - 5 sin (x) - 15 cos (x)

= - 5 sin (x) - 15 cos (x)

= \frac{- 5 sin ( x ) - 15 cos ( x )}{3 sin ( x ) + 4 cos ( x )}

\frac{- 5 sin ( x ) - 15 cos ( x )}{3 sin ( x ) + 4 cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 5 sin (x) - 15 cos (x) = 0

Rewrite using trig identities

- 5 sin (x) - 15 cos (x) = 0

Divide both sides by

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

\frac{- 5 sin ( x ) - 15 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

- \frac{5 sin ( x )}{cos ( x )} - 15 = 0

Use the basic trigonometric identity:

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

- 5 tan (x) - 15 = 0

- 5 tan (x) - 15 = 0

Move

15

to the right side

- 5 tan (x) - 15 = 0

Add

15

to both sides

- 5 tan (x) - 15 + 15 = 0 + 15

Simplify

- 5 tan (x) = 15

- 5 tan (x) = 15

Divide both sides by

- 5

- 5 tan (x) = 15

Divide both sides by

- 5

\frac{- 5 tan ( x )}{- 5} = \frac{15}{- 5}

Simplify

\frac{- 5 tan ( x )}{- 5} = \frac{15}{- 5}

Simplify

\frac{- 5 tan ( x )}{- 5} : tan (x)

\frac{- 5 tan ( x )}{- 5}

Apply the fraction rule:

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

= \frac{5 tan ( x )}{5}

Divide the numbers:

\frac{5}{5} = 1

= tan (x)

Simplify

\frac{15}{- 5} : - 3

\frac{15}{- 5}

Apply the fraction rule:

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

= - \frac{15}{5}

Divide the numbers:

\frac{15}{5} = 3

= - 3

tan (x) = - 3

tan (x) = - 3

tan (x) = - 3

Apply trig inverse properties

tan (x) = - 3

General solutions for

tan (x) = - 3

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

x = arctan (- 3) + πn

x = arctan (- 3) + πn

Show solutions in decimal form

x = - 1.24904 \dots + πn

Graph

Popular Examples

(1+tan(t))/(sin(t))=0

Frequently Asked Questions (FAQ)

What is the general solution for (4sin(x)-3cos(x))/(3sin(x)+4cos(x))=3 ?
The general solution for (4sin(x)-3cos(x))/(3sin(x)+4cos(x))=3 is x=-1.24904…+pin