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

The general solution for (3sin(x)-cos(x))/(sin(x)+cos(x))=4 is x=-1.37340…+pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

Solution

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

Solution

x = - 1.37340 \dots + πn

Degrees

x = - 78.69006 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

4

from both sides

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

Simplify

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Expand

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

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

Expand

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

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

Apply the distributive law:

a (b + c) = ab + a c

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

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

Apply minus-plus rules

+ (- a) = - a

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

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

Simplify

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

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

Add similar elements:

- cos (x) - 4 cos (x) = - 5 cos (x)

= 3 sin (x) - 5 cos (x) - 4 sin (x)

Add similar elements:

3 sin (x) - 4 sin (x) = - sin (x)

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

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

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

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

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

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

- tan (x) - 5 = 0

- tan (x) - 5 = 0

Move

5

to the right side

- tan (x) - 5 = 0

Add

5

to both sides

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

Simplify

- tan (x) = 5

- tan (x) = 5

Divide both sides by

- 1

- tan (x) = 5

Divide both sides by

- 1

\frac{- tan ( x )}{- 1} = \frac{5}{- 1}

Simplify

tan (x) = - 5

tan (x) = - 5

Apply trig inverse properties

tan (x) = - 5

General solutions for

tan (x) = - 5

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

x = arctan (- 5) + πn

x = arctan (- 5) + πn

Show solutions in decimal form

x = - 1.37340 \dots + πn

Popular Examples

1/(e^{sin(x))}=1

\frac{1}{e ^{s i n (x)}} = 1

|cos(x)|=(sqrt(3))/2

∣ cos (x) ∣ = \frac{3}{2}

tan(θ)=(-1)/(sqrt(3)),0<= θ<= 360

tan (θ) = \frac{- 1}{3}, 0^{\circ} \leq θ \leq 36 0^{\circ}

4cos(pi/4 (x+3))=0

4 cos (\frac{π}{4} (x + 3)) = 0

sin^3(x)=cos(x)

sin^{3} (x) = cos (x)

Frequently Asked Questions (FAQ)

What is the general solution for (3sin(x)-cos(x))/(sin(x)+cos(x))=4 ?
The general solution for (3sin(x)-cos(x))/(sin(x)+cos(x))=4 is x=-1.37340…+pin