What is the general solution for tan(θ+pi)+2sin(θ+pi)=0 ?

The general solution for tan(θ+pi)+2sin(θ+pi)=0 is θ=2pin,θ=pi+2pin,θ= pi/3+2pin,θ=(5pi)/3+2pin

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Popular >

tan(θ+pi)+2sin(θ+pi)=0

Solution

tan (θ + π) + 2 sin (θ + π) = 0

Solution

θ = 2 πn, θ = π + 2 πn, θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

Degrees

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n, θ = 6 0^{\circ} + 36 0^{\circ} n, θ = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

tan (θ + π) + 2 sin (θ + π) = 0

Rewrite using trig identities

tan (θ + π) + 2 sin (θ + π) = 0

Rewrite using trig identities

tan (θ + π)

Use the basic trigonometric identity:

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

= \frac{sin ( θ + π )}{cos ( θ + π )}

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= \frac{sin ( θ ) cos ( π ) + cos ( θ ) sin ( π )}{cos ( θ + π )}

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= \frac{sin ( θ ) cos ( π ) + cos ( θ ) sin ( π )}{cos ( θ ) cos ( π ) - sin ( θ ) sin ( π )}

Simplify

\frac{sin ( θ ) cos ( π ) + cos ( θ ) sin ( π )}{cos ( θ ) cos ( π ) - sin ( θ ) sin ( π )} : \frac{sin ( θ )}{cos ( θ )}

\frac{sin ( θ ) cos ( π ) + cos ( θ ) sin ( π )}{cos ( θ ) cos ( π ) - sin ( θ ) sin ( π )}

sin (θ) cos (π) + cos (θ) sin (π) = - sin (θ)

sin (θ) cos (π) + cos (θ) sin (π)

sin (θ) cos (π) = - sin (θ)

sin (θ) cos (π)

Simplify

cos (π) : - 1

cos (π)

Use the following trivial identity:

cos (π) = (- 1)

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}

= - 1

= (- 1) sin (θ)

Refine

= - sin (θ)

= - sin (θ) + sin (π) cos (θ)

cos (θ) sin (π) = 0

cos (θ) sin (π)

Simplify

sin (π) : 0

sin (π)

Use the following trivial identity:

sin (π) = 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}

= 0

= 0 \cdot cos (θ)

Apply rule

0 \cdot a = 0

= 0

= - sin (θ) + 0

- sin (θ) + 0 = - sin (θ)

= - sin (θ)

= \frac{- sin ( θ )}{cos ( π ) cos ( θ ) - sin ( π ) sin ( θ )}

cos (θ) cos (π) - sin (θ) sin (π) = - cos (θ)

cos (θ) cos (π) - sin (θ) sin (π)

cos (θ) cos (π) = - cos (θ)

cos (θ) cos (π)

Simplify

cos (π) : - 1

cos (π)

Use the following trivial identity:

cos (π) = (- 1)

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}

= - 1

= (- 1) cos (θ)

Refine

= - cos (θ)

= - cos (θ) - sin (π) sin (θ)

sin (θ) sin (π) = 0

sin (θ) sin (π)

Simplify

sin (π) : 0

sin (π)

Use the following trivial identity:

sin (π) = 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}

= 0

= 0 \cdot sin (θ)

Apply rule

0 \cdot a = 0

= 0

= - cos (θ) - 0

- cos (θ) - 0 = - cos (θ)

= - cos (θ)

= \frac{- sin ( θ )}{- cos ( θ )}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{sin ( θ )}{cos ( θ )}

= \frac{sin ( θ )}{cos ( θ )}

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (θ) cos (π) + cos (θ) sin (π)

Simplify

sin (θ) cos (π) + cos (θ) sin (π) : - sin (θ)

sin (θ) cos (π) + cos (θ) sin (π)

sin (θ) cos (π) = - sin (θ)

sin (θ) cos (π)

Simplify

cos (π) : - 1

cos (π)

Use the following trivial identity:

cos (π) = (- 1)

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}

= - 1

= (- 1) sin (θ)

Refine

= - sin (θ)

= - sin (θ) + sin (π) cos (θ)

cos (θ) sin (π) = 0

cos (θ) sin (π)

Simplify

sin (π) : 0

sin (π)

Use the following trivial identity:

sin (π) = 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}

= 0

= 0 \cdot cos (θ)

Apply rule

0 \cdot a = 0

= 0

= - sin (θ) + 0

- sin (θ) + 0 = - sin (θ)

= - sin (θ)

= - sin (θ)

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

Simplify

\frac{sin ( θ )}{cos ( θ )} + 2 (- sin (θ)) : \frac{sin ( θ )}{cos ( θ )} - 2 sin (θ)

\frac{sin ( θ )}{cos ( θ )} + 2 (- sin (θ))

Remove parentheses:

(- a) = - a

= \frac{sin ( θ )}{cos ( θ )} - 2 sin (θ)

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

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

Simplify

\frac{sin ( θ )}{cos ( θ )} - 2 sin (θ) : \frac{sin ( θ ) - 2 sin ( θ ) cos ( θ )}{cos ( θ )}

\frac{sin ( θ )}{cos ( θ )} - 2 sin (θ)

Convert element to fraction:

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

= \frac{sin ( θ )}{cos ( θ )} - \frac{2 sin ( θ ) cos ( θ )}{cos ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ( θ ) - 2 sin ( θ ) cos ( θ )}{cos ( θ )}

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

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

sin (θ) - 2 sin (θ) cos (θ) = 0

Factor

sin (θ) - 2 sin (θ) cos (θ) : - sin (θ) (2 cos (θ) - 1)

sin (θ) - 2 sin (θ) cos (θ)

Factor out common term

- sin (θ)

= - sin (θ) (- 1 + 2 cos (θ))

- sin (θ) (2 cos (θ) - 1) = 0

Solving each part separately

sin (θ) = 0 or 2 cos (θ) - 1 = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

sin (θ) = 0

General solutions for

sin (θ) = 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}

θ = 0 + 2 πn, θ = π + 2 πn

θ = 0 + 2 πn, θ = π + 2 πn

Solve

θ = 0 + 2 πn : θ = 2 πn

θ = 0 + 2 πn

0 + 2 πn = 2 πn

θ = 2 πn

θ = 2 πn, θ = π + 2 πn

2 cos (θ) - 1 = 0 : θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

2 cos (θ) - 1 = 0

Move

1

to the right side

2 cos (θ) - 1 = 0

Add

1

to both sides

2 cos (θ) - 1 + 1 = 0 + 1

Simplify

2 cos (θ) = 1

2 cos (θ) = 1

Divide both sides by

2

2 cos (θ) = 1

Divide both sides by

2

\frac{2 cos ( θ )}{2} = \frac{1}{2}

Simplify

cos (θ) = \frac{1}{2}

cos (θ) = \frac{1}{2}

General solutions for

cos (θ) = \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}

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

Combine all the solutions

θ = 2 πn, θ = π + 2 πn, θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

Popular Examples

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sin (θ) = \frac{2}{3}, \frac{π}{2} < θ < π, cos (2 θ) = x

tan(3x)=2

tan (3 x) = 2

tan(θ)=sqrt(3)-pi

tan (θ) = 3 - π

sin(x)= 15/39

sin (x) = \frac{15}{39}

tan(a)=(40)/(12.5)

tan (a) = \frac{40}{12.5}

Frequently Asked Questions (FAQ)

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