What is the general solution for (cos(x)-3sin(x))/(sin(x)+3cos(x))=-1/2 ?

The general solution for (cos(x)-3sin(x))/(sin(x)+3cos(x))=-1/2 is x= pi/4+pin

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Popular >

(cos(x)-3sin(x))/(sin(x)+3cos(x))=-1/2

Solution

\frac{cos ( x ) - 3 sin ( x )}{sin ( x ) + 3 cos ( x )} = - \frac{1}{2}

Solution

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

\frac{cos ( x ) - 3 sin ( x )}{sin ( x ) + 3 cos ( x )} = - \frac{1}{2}

Subtract

- \frac{1}{2}

from both sides

\frac{cos ( x ) - 3 sin ( x )}{sin ( x ) + 3 cos ( x )} + \frac{1}{2} = 0

Simplify

\frac{cos ( x ) - 3 sin ( x )}{sin ( x ) + 3 cos ( x )} + \frac{1}{2} : \frac{5 cos ( x ) - 5 sin ( x )}{2 ( sin ( x ) + 3 cos ( x ) )}

\frac{cos ( x ) - 3 sin ( x )}{sin ( x ) + 3 cos ( x )} + \frac{1}{2}

Least Common Multiplier of

sin (x) + 3 cos (x), 2 : 2 (sin (x) + 3 cos (x))

sin (x) + 3 cos (x), 2

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (x) + 3 cos (x)

2

= 2 (sin (x) + 3 cos (x))

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

2 (sin (x) + 3 cos (x))

For

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

multiply the denominator and numerator by

2

\frac{cos ( x ) - 3 sin ( x )}{sin ( x ) + 3 cos ( x )} = \frac{( cos ( x ) - 3 sin ( x ) ) \cdot 2}{( sin ( x ) + 3 cos ( x ) ) \cdot 2}

For

\frac{1}{2} :

multiply the denominator and numerator by

sin (x) + 3 cos (x)

\frac{1}{2} = \frac{1 \cdot ( sin ( x ) + 3 cos ( x ) )}{2 ( sin ( x ) + 3 cos ( x ) )} = \frac{sin ( x ) + 3 cos ( x )}{2 ( sin ( x ) + 3 cos ( x ) )}

= \frac{( cos ( x ) - 3 sin ( x ) ) \cdot 2}{( sin ( x ) + 3 cos ( x ) ) \cdot 2} + \frac{sin ( x ) + 3 cos ( x )}{2 ( sin ( x ) + 3 cos ( x ) )}

Since the denominators are equal, combine the fractions:

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

= \frac{( cos ( x ) - 3 sin ( x ) ) \cdot 2 + sin ( x ) + 3 cos ( x )}{2 ( sin ( x ) + 3 cos ( x ) )}

Expand

(cos (x) - 3 sin (x)) \cdot 2 + sin (x) + 3 cos (x) : 5 cos (x) - 5 sin (x)

(cos (x) - 3 sin (x)) \cdot 2 + sin (x) + 3 cos (x)

= 2 (cos (x) - 3 sin (x)) + sin (x) + 3 cos (x)

Expand

2 (cos (x) - 3 sin (x)) : 2 cos (x) - 6 sin (x)

2 (cos (x) - 3 sin (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = cos (x), c = 3 sin (x)

= 2 cos (x) - 2 \cdot 3 sin (x)

Multiply the numbers:

2 \cdot 3 = 6

= 2 cos (x) - 6 sin (x)

= 2 cos (x) - 6 sin (x) + sin (x) + 3 cos (x)

Simplify

2 cos (x) - 6 sin (x) + sin (x) + 3 cos (x) : 5 cos (x) - 5 sin (x)

2 cos (x) - 6 sin (x) + sin (x) + 3 cos (x)

Add similar elements:

2 cos (x) + 3 cos (x) = 5 cos (x)

= 5 cos (x) - 6 sin (x) + sin (x)

Add similar elements:

- 6 sin (x) + sin (x) = - 5 sin (x)

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

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

= \frac{5 cos ( x ) - 5 sin ( x )}{2 ( sin ( x ) + 3 cos ( x ) )}

\frac{5 cos ( x ) - 5 sin ( x )}{2 ( sin ( x ) + 3 cos ( x ) )} = 0

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

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

5 - 5 tan (x) = 0

5 - 5 tan (x) = 0

Move

5

to the right side

5 - 5 tan (x) = 0

Subtract

5

from both sides

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

Simplify

- 5 tan (x) = - 5

- 5 tan (x) = - 5

Divide both sides by

- 5

- 5 tan (x) = - 5

Divide both sides by

- 5

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

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

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}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Popular Examples

cos(pi/2-x)=0

cos (\frac{π}{2} - x) = 0

1sin(45)=2.42sin(r)

1 sin (4 5^{\circ}) = 2.42 sin (r)

sec(x)=4sin(x)

sec (x) = 4 sin (x)

sinh(npi)=0

sinh (nπ) = 0

solvefor x,z=sin(xy^2)solve for

x, z = sin (x y^{2})

Frequently Asked Questions (FAQ)

What is the general solution for (cos(x)-3sin(x))/(sin(x)+3cos(x))=-1/2 ?
The general solution for (cos(x)-3sin(x))/(sin(x)+3cos(x))=-1/2 is x= pi/4+pin