What is the general solution for 2sin(x)+sqrt(3)tan(x)=0 ?

The general solution for 2sin(x)+sqrt(3)tan(x)=0 is x=2pin,x=pi+2pin,x=(5pi)/6+2pin,x=(7pi)/6+2pin

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Popular >

2sin(x)+sqrt(3)tan(x)=0

Solution

2 sin (x) + 3 tan (x) = 0

Solution

x = 2 πn, x = π + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sin (x) + 3 tan (x) = 0

Express with sin, cos

2 sin (x) + 3 tan (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply

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

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

Multiply fractions:

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

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

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

Convert element to fraction:

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

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

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

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

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

sin (x) 3 + 2 cos (x) sin (x) = 0

Factor

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

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

Factor out common term

sin (x)

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

sin (x) (3 + 2 cos (x)) = 0

Solving each part separately

sin (x) = 0 or 3 + 2 cos (x) = 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

3 + 2 cos (x) = 0 : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

3 + 2 cos (x) = 0

Move

3

to the right side

3 + 2 cos (x) = 0

Subtract

3

from both sides

3 + 2 cos (x) - 3 = 0 - 3

Simplify

2 cos (x) = - 3

2 cos (x) = - 3

Divide both sides by

2

2 cos (x) = - 3

Divide both sides by

2

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

Simplify

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

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

General solutions for

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

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

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Popular Examples

tan(t)=2,0<t< pi/2 ,cos(t)

tan (t) = 2, 0 < t < \frac{π}{2}, cos (t)

(cos(x)+sin(x))/(sin(x))= 1/(tan(x))

\frac{cos ( x ) + sin ( x )}{sin ( x )} = \frac{1}{tan ( x )}

cos(2x)-3sin(-x)-2=0

cos (2 x) - 3 sin (- x) - 2 = 0

sin(θ)= 1/4 ,0<θ< pi/2

sin (θ) = \frac{1}{4}, 0 < θ < \frac{π}{2}

sin(2θ)=(1.35*10)/((38)^2)

sin (2 θ) = \frac{1.35 \cdot 10}{( 38 ) ^{2}}

Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x)+sqrt(3)tan(x)=0 ?
The general solution for 2sin(x)+sqrt(3)tan(x)=0 is x=2pin,x=pi+2pin,x=(5pi)/6+2pin,x=(7pi)/6+2pin