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Popular Trigonometry >

((tan(x)+2sin(x)))/((tan(x)-2sin(x)))=3

Solution

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

Solution

x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

Degrees

x = 75.52248 \dots^{\circ} + 36 0^{\circ} n, x = 284.47751 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

3

from both sides

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

Simplify

\frac{tan ( x ) + 2 sin ( x )}{tan ( x ) - 2 sin ( x )} - 3 : \frac{- 2 tan ( x ) + 8 sin ( x )}{tan ( x ) - 2 sin ( x )}

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

Convert element to fraction:

3 = \frac{3 ( tan ( x ) - 2 sin ( x ) )}{tan ( x ) - 2 sin ( x )}

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

Since the denominators are equal, combine the fractions:

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

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

Expand

tan (x) + 2 sin (x) - 3 (tan (x) - 2 sin (x)) : - 2 tan (x) + 8 sin (x)

tan (x) + 2 sin (x) - 3 (tan (x) - 2 sin (x))

Expand

- 3 (tan (x) - 2 sin (x)) : - 3 tan (x) + 6 sin (x)

- 3 (tan (x) - 2 sin (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 3, b = tan (x), c = 2 sin (x)

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

Apply minus-plus rules

- (- a) = a

= - 3 tan (x) + 3 \cdot 2 sin (x)

Multiply the numbers:

3 \cdot 2 = 6

= - 3 tan (x) + 6 sin (x)

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

Simplify

tan (x) + 2 sin (x) - 3 tan (x) + 6 sin (x) : - 2 tan (x) + 8 sin (x)

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

Add similar elements:

2 sin (x) + 6 sin (x) = 8 sin (x)

= tan (x) + 8 sin (x) - 3 tan (x)

Add similar elements:

tan (x) - 3 tan (x) = - 2 tan (x)

= - 2 tan (x) + 8 sin (x)

= - 2 tan (x) + 8 sin (x)

= \frac{- 2 tan ( x ) + 8 sin ( x )}{tan ( x ) - 2 sin ( x )}

\frac{- 2 tan ( x ) + 8 sin ( x )}{tan ( x ) - 2 sin ( x )} = 0

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

- 2 tan (x) + 8 sin (x) = 0

Express with sin, cos

- 2 tan (x) + 8 sin (x)

Use the basic trigonometric identity:

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

= - 2 \cdot \frac{sin ( x )}{cos ( x )} + 8 sin (x)

Simplify

- 2 \cdot \frac{sin ( x )}{cos ( x )} + 8 sin (x) : \frac{- 2 sin ( x ) + 8 sin ( x ) cos ( x )}{cos ( x )}

- 2 \cdot \frac{sin ( x )}{cos ( x )} + 8 sin (x)

Multiply

2 \cdot \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( x )}{cos ( x )}

2 \cdot \frac{sin ( x )}{cos ( x )}

Multiply fractions:

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

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

= - \frac{2 sin ( x )}{cos ( x )} + 8 sin (x)

Convert element to fraction:

8 sin (x) = \frac{8 sin ( x ) cos ( x )}{cos ( x )}

= - \frac{sin ( x ) \cdot 2}{cos ( x )} + \frac{8 sin ( x ) cos ( x )}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{- sin ( x ) \cdot 2 + 8 sin ( x ) cos ( x )}{cos ( x )}

= \frac{- 2 sin ( x ) + 8 sin ( x ) cos ( x )}{cos ( x )}

\frac{- 2 sin ( x ) + 8 cos ( x ) sin ( x )}{cos ( x )} = 0

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

- 2 sin (x) + 8 cos (x) sin (x) = 0

Factor

- 2 sin (x) + 8 cos (x) sin (x) : 2 sin (x) (4 cos (x) - 1)

- 2 sin (x) + 8 cos (x) sin (x)

Rewrite

8

4 \cdot 2

= - 2 sin (x) + 4 \cdot 2 sin (x) cos (x)

Factor out common term

2 sin (x)

= 2 sin (x) (- 1 + 4 cos (x))

2 sin (x) (4 cos (x) - 1) = 0

Solving each part separately

sin (x) = 0 or 4 cos (x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

4 cos (x) - 1 = 0 : x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

4 cos (x) - 1 = 0

Move

1

to the right side

4 cos (x) - 1 = 0

Add

1

to both sides

4 cos (x) - 1 + 1 = 0 + 1

Simplify

4 cos (x) = 1

4 cos (x) = 1

Divide both sides by

4

4 cos (x) = 1

Divide both sides by

4

\frac{4 cos ( x )}{4} = \frac{1}{4}

Simplify

cos (x) = \frac{1}{4}

cos (x) = \frac{1}{4}

Apply trig inverse properties

cos (x) = \frac{1}{4}

General solutions for

cos (x) = \frac{1}{4}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

Since the equation is undefined for:

2 πn, π + 2 πn

x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

Show solutions in decimal form

x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

Graph

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