English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

tan(2x)=5sin(2x)

Solution

tan (2 x) = 5 sin (2 x)

Solution

x = πn, x = \frac{π}{2} + πn, x = \frac{1.36943 \dots}{2} + πn, x = π - \frac{1.36943 \dots}{2} + πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n, x = 39.23152 \dots^{\circ} + 18 0^{\circ} n, x = 140.76847 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan (2 x) = 5 sin (2 x)

Subtract

5 sin (2 x)

from both sides

tan (2 x) - 5 sin (2 x) = 0

Express with sin, cos

tan (2 x) - 5 sin (2 x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Convert element to fraction:

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

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

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

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

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

sin (2 x) - 5 cos (2 x) sin (2 x) = 0

Factor

sin (2 x) - 5 cos (2 x) sin (2 x) : - sin (2 x) (5 cos (2 x) - 1)

sin (2 x) - 5 cos (2 x) sin (2 x)

Factor out common term

- sin (2 x)

= - sin (2 x) (- 1 + 5 cos (2 x))

- sin (2 x) (5 cos (2 x) - 1) = 0

Solving each part separately

sin (2 x) = 0 or 5 cos (2 x) - 1 = 0

sin (2 x) = 0 : x = πn, x = \frac{π}{2} + πn

sin (2 x) = 0

General solutions for

sin (2 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}

2 x = 0 + 2 πn, 2 x = π + 2 πn

2 x = 0 + 2 πn, 2 x = π + 2 πn

Solve

2 x = 0 + 2 πn : x = πn

2 x = 0 + 2 πn

0 + 2 πn = 2 πn

2 x = 2 πn

Divide both sides by

2

2 x = 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 πn}{2}

Simplify

x = πn

x = πn

Solve

2 x = π + 2 πn : x = \frac{π}{2} + πn

2 x = π + 2 πn

Divide both sides by

2

2 x = π + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = πn, x = \frac{π}{2} + πn

5 cos (2 x) - 1 = 0 : x = \frac{arccos ( \frac{1}{5} )}{2} + πn, x = π - \frac{arccos ( \frac{1}{5} )}{2} + πn

5 cos (2 x) - 1 = 0

Move

1

to the right side

5 cos (2 x) - 1 = 0

Add

1

to both sides

5 cos (2 x) - 1 + 1 = 0 + 1

Simplify

5 cos (2 x) = 1

5 cos (2 x) = 1

Divide both sides by

5

5 cos (2 x) = 1

Divide both sides by

5

\frac{5 cos ( 2 x )}{5} = \frac{1}{5}

Simplify

cos (2 x) = \frac{1}{5}

cos (2 x) = \frac{1}{5}

Apply trig inverse properties

cos (2 x) = \frac{1}{5}

General solutions for

cos (2 x) = \frac{1}{5}

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

2 x = arccos (\frac{1}{5}) + 2 πn, 2 x = 2 π - arccos (\frac{1}{5}) + 2 πn

2 x = arccos (\frac{1}{5}) + 2 πn, 2 x = 2 π - arccos (\frac{1}{5}) + 2 πn

Solve

2 x = arccos (\frac{1}{5}) + 2 πn : x = \frac{arccos ( \frac{1}{5} )}{2} + πn

2 x = arccos (\frac{1}{5}) + 2 πn

Divide both sides by

2

2 x = arccos (\frac{1}{5}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{arccos ( \frac{1}{5} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{arccos ( \frac{1}{5} )}{2} + πn

x = \frac{arccos ( \frac{1}{5} )}{2} + πn

Solve

2 x = 2 π - arccos (\frac{1}{5}) + 2 πn : x = π - \frac{arccos ( \frac{1}{5} )}{2} + πn

2 x = 2 π - arccos (\frac{1}{5}) + 2 πn

Divide both sides by

2

2 x = 2 π - arccos (\frac{1}{5}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 π}{2} - \frac{arccos ( \frac{1}{5} )}{2} + \frac{2 πn}{2}

Simplify

x = π - \frac{arccos ( \frac{1}{5} )}{2} + πn

x = π - \frac{arccos ( \frac{1}{5} )}{2} + πn

x = \frac{arccos ( \frac{1}{5} )}{2} + πn, x = π - \frac{arccos ( \frac{1}{5} )}{2} + πn

Combine all the solutions

x = πn, x = \frac{π}{2} + πn, x = \frac{arccos ( \frac{1}{5} )}{2} + πn, x = π - \frac{arccos ( \frac{1}{5} )}{2} + πn

Show solutions in decimal form

x = πn, x = \frac{π}{2} + πn, x = \frac{1.36943 \dots}{2} + πn, x = π - \frac{1.36943 \dots}{2} + πn

Popular Examples

2sin^2(x)-7sin(x)-4=0

2 sin^{2} (x) - 7 sin (x) - 4 = 0

sqrt(2)cos(2θ)+1=0

2 cos (2 θ) + 1 = 0

(sqrt(2)sin(x))+1=0

(2 sin (x)) + 1 = 0

sin(x)+2=3

sin (x) + 2 = 3

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

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