What is the general solution for sin(x)=(-tan(x))/2 ?

The general solution for sin(x)=(-tan(x))/2 is x=2pin,x=pi+2pin,x=(2pi)/3+2pin,x=(4pi)/3+2pin

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Popular >

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

Solution

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

Solution

x = 2 πn, x = π + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

\frac{- tan ( x )}{2}

from both sides

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

Simplify

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

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

Convert element to fraction:

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

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

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 + tan ( x )}{2}

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

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

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

Express with sin, cos

tan (x) + 2 sin (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Convert element to fraction:

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

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

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

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

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

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

Factor

sin (x) + 2 cos (x) sin (x) : sin (x) (2 cos (x) + 1)

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

Factor out common term

sin (x)

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

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

Solving each part separately

sin (x) = 0 or 2 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

2 cos (x) + 1 = 0 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

2 cos (x) + 1 = 0

Move

1

to the right side

2 cos (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

2 cos (x) = - 1

2 cos (x) = - 1

Divide both sides by

2

2 cos (x) = - 1

Divide both sides by

2

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

Simplify

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

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

General solutions for

cos (x) = - \frac{1}{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{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Popular Examples

2sin(3θ)-1=0,0<= θ<= 2pi

2 sin (3 θ) - 1 = 0, 0 \leq θ \leq 2 π

sin(θ)=-5/7

sin (θ) = - \frac{5}{7}

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

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

sec^2(x)-1=1

sec^{2} (x) - 1 = 1

sec(θ)=8

sec (θ) = 8

Frequently Asked Questions (FAQ)

What is the general solution for sin(x)=(-tan(x))/2 ?
The general solution for sin(x)=(-tan(x))/2 is x=2pin,x=pi+2pin,x=(2pi)/3+2pin,x=(4pi)/3+2pin