What is the general solution for (sin(x))/(cos(x))-5+(4cos(x))/(sin(x))=0 ?

The general solution for (sin(x))/(cos(x))-5+(4cos(x))/(sin(x))=0 is x= pi/4+pin,x=1.32581…+pin

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Popular Trigonometry >

(sin(x))/(cos(x))-5+(4cos(x))/(sin(x))=0

Solution

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

Solution

x = \frac{π}{4} + πn, x = 1.32581 \dots + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n, x = 75.96375 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Simplify

\frac{sin ( x )}{cos ( x )} - 5 + \frac{4 cos ( x )}{sin ( x )} : \frac{sin ^{2} ( x ) - 5 cos ( x ) sin ( x ) + 4 cos ^{2} ( x )}{cos ( x ) sin ( x )}

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

Convert element to fraction:

5 = \frac{5}{1}

= \frac{sin ( x )}{cos ( x )} - \frac{5}{1} + \frac{4 cos ( x )}{sin ( x )}

Least Common Multiplier of

cos (x), 1, sin (x) : cos (x) sin (x)

cos (x), 1, sin (x)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear in at least one of the factored expressions

= cos (x) sin (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

cos (x) sin (x)

For

\frac{sin ( x )}{cos ( x )} :

multiply the denominator and numerator by

sin (x)

\frac{sin ( x )}{cos ( x )} = \frac{sin ( x ) sin ( x )}{cos ( x ) sin ( x )} = \frac{sin ^{2} ( x )}{cos ( x ) sin ( x )}

For

\frac{5}{1} :

multiply the denominator and numerator by

cos (x) sin (x)

\frac{5}{1} = \frac{5 cos ( x ) sin ( x )}{1 \cdot cos ( x ) sin ( x )} = \frac{5 cos ( x ) sin ( x )}{cos ( x ) sin ( x )}

For

\frac{4 cos ( x )}{sin ( x )} :

multiply the denominator and numerator by

cos (x)

\frac{4 cos ( x )}{sin ( x )} = \frac{4 cos ( x ) cos ( x )}{sin ( x ) cos ( x )} = \frac{4 cos ^{2} ( x )}{cos ( x ) sin ( x )}

= \frac{sin ^{2} ( x )}{cos ( x ) sin ( x )} - \frac{5 cos ( x ) sin ( x )}{cos ( x ) sin ( x )} + \frac{4 cos ^{2} ( x )}{cos ( x ) sin ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ^{2} ( x ) - 5 cos ( x ) sin ( x ) + 4 cos ^{2} ( x )}{cos ( x ) sin ( x )}

\frac{sin ^{2} ( x ) - 5 cos ( x ) sin ( x ) + 4 cos ^{2} ( x )}{cos ( x ) sin ( x )} = 0

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

sin^{2} (x) - 5 cos (x) sin (x) + 4 cos^{2} (x) = 0

Factor

sin^{2} (x) - 5 cos (x) sin (x) + 4 cos^{2} (x) : (sin (x) - cos (x)) (sin (x) - 4 cos (x))

sin^{2} (x) - 5 cos (x) sin (x) + 4 cos^{2} (x)

Break the expression into groups

sin^{2} (x) - 5 sin (x) cos (x) + 4 cos^{2} (x)

Definition

Factors of

4 : 1, 2, 4

4

Divisors (Factors)

Find the Prime factors of

4 : 2, 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2

Add the prime factors:

2

Add 1 and the number

4

itself

1, 4

The factors of

4

1, 2, 4

Negative factors of

4 : - 1, - 2, - 4

Multiply the factors by

- 1

to get the negative factors

- 1, - 2, - 4

For every two factors such that

u * v = 4,

check if

u + v = - 5

Check

u = 1, v = 4 : u * v = 4, u + v = 5 \Rightarrow

FalseCheck

u = 2, v = 2 : u * v = 4, u + v = 4 \Rightarrow

False

u = - 1, v = - 4

Group into

(a x^{2} + ux y) + (vx y + c y^{2})

(sin^{2} (x) - sin (x) cos (x)) + (- 4 sin (x) cos (x) + 4 cos^{2} (x))

= (sin^{2} (x) - sin (x) cos (x)) + (- 4 sin (x) cos (x) + 4 cos^{2} (x))

Factor out

sin (x)

from

sin^{2} (x) - sin (x) cos (x) : sin (x) (sin (x) - cos (x))

sin^{2} (x) - sin (x) cos (x)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

sin^{2} (x) = sin (x) sin (x)

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

Factor out common term

sin (x)

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

Factor out

- 4 cos (x)

from

- 4 sin (x) cos (x) + 4 cos^{2} (x) : - 4 cos (x) (sin (x) - cos (x))

- 4 sin (x) cos (x) + 4 cos^{2} (x)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

cos^{2} (x) = cos (x) cos (x)

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

Factor out common term

- 4 cos (x)

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

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

Factor out common term

sin (x) - cos (x)

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

(sin (x) - cos (x)) (sin (x) - 4 cos (x)) = 0

Solving each part separately

sin (x) - cos (x) = 0 or sin (x) - 4 cos (x) = 0

sin (x) - cos (x) = 0 : x = \frac{π}{4} + πn

sin (x) - cos (x) = 0

Rewrite using trig identities

sin (x) - cos (x) = 0

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

tan (x) - 1 = 0

tan (x) - 1 = 0

Move

1

to the right side

tan (x) - 1 = 0

Add

1

to both sides

tan (x) - 1 + 1 = 0 + 1

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

sin (x) - 4 cos (x) = 0 : x = arctan (4) + πn

sin (x) - 4 cos (x) = 0

Rewrite using trig identities

sin (x) - 4 cos (x) = 0

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

tan (x) - 4 = 0

tan (x) - 4 = 0

Move

4

to the right side

tan (x) - 4 = 0

Add

4

to both sides

tan (x) - 4 + 4 = 0 + 4

Simplify

tan (x) = 4

tan (x) = 4

Apply trig inverse properties

tan (x) = 4

General solutions for

tan (x) = 4

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

x = arctan (4) + πn

x = arctan (4) + πn

Combine all the solutions

x = \frac{π}{4} + πn, x = arctan (4) + πn

Show solutions in decimal form

x = \frac{π}{4} + πn, x = 1.32581 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for (sin(x))/(cos(x))-5+(4cos(x))/(sin(x))=0 ?
The general solution for (sin(x))/(cos(x))-5+(4cos(x))/(sin(x))=0 is x= pi/4+pin,x=1.32581…+pin