What is the general solution for 3tan(x)=sin(x) ?

The general solution for 3tan(x)=sin(x) is x=2pin,x=pi+2pin

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Popular >

3tan(x)=sin(x)

Solution

3 tan (x) = sin (x)

Solution

x = 2 πn, x = π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

3 tan (x) = sin (x)

Subtract

sin (x)

from both sides

3 tan (x) - sin (x) = 0

Express with sin, cos

- sin (x) + 3 tan (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply

3 \cdot \frac{sin ( x )}{cos ( x )} : \frac{3 sin ( x )}{cos ( x )}

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

Multiply fractions:

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

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

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

Convert element to fraction:

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

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

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

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

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

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

Factor

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

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

Factor out common term

- sin (x)

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

- sin (x) (cos (x) - 3) = 0

Solving each part separately

sin (x) = 0 or cos (x) - 3 = 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

cos (x) - 3 = 0 :

No Solution

cos (x) - 3 = 0

Move

3

to the right side

cos (x) - 3 = 0

Add

3

to both sides

cos (x) - 3 + 3 = 0 + 3

Simplify

cos (x) = 3

cos (x) = 3

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

Popular Examples

5=sinh(x)

5 = sinh (x)

-16cos(2x)=0

- 16 cos (2 x) = 0

6sqrt(2)+12cos(x-45)=0

62 + 12 cos (x - 4 5^{\circ}) = 0

(2*9.8*l(1-cos(θ)))=0.018

(2 \cdot 9.8 \cdot l (1 - cos (θ))) = 0.018

1/(tan(x))=0.5

\frac{1}{tan ( x )} = 0.5

Frequently Asked Questions (FAQ)

What is the general solution for 3tan(x)=sin(x) ?
The general solution for 3tan(x)=sin(x) is x=2pin,x=pi+2pin