What is the general solution for tan(θ)=-sqrt(2)sin(θ) ?

The general solution for tan(θ)=-sqrt(2)sin(θ) is θ=2pin,θ=pi+2pin,θ=(3pi)/4+2pin,θ=(5pi)/4+2pin

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tan(θ)=-sqrt(2)sin(θ)

Solution

tan (θ) = - 2 sin (θ)

Solution

θ = 2 πn, θ = π + 2 πn, θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

Degrees

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n, θ = 13 5^{\circ} + 36 0^{\circ} n, θ = 22 5^{\circ} + 36 0^{\circ} n

Solution steps

tan (θ) = - 2 sin (θ)

Subtract

- 2 sin (θ)

from both sides

tan (θ) + 2 sin (θ) = 0

Express with sin, cos

tan (θ) + sin (θ) 2

Use the basic trigonometric identity:

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

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

Simplify

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

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

Convert element to fraction:

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

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

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

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

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

sin (θ) + cos (θ) sin (θ) 2 = 0

Factor

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

sin (θ) + cos (θ) sin (θ) 2

Factor out common term

sin (θ)

= sin (θ) (1 + cos (θ) 2)

sin (θ) (1 + 2 cos (θ)) = 0

Solving each part separately

sin (θ) = 0 or 1 + 2 cos (θ) = 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

1 + 2 cos (θ) = 0 : θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

1 + 2 cos (θ) = 0

Move

1

to the right side

1 + 2 cos (θ) = 0

Subtract

1

from both sides

1 + 2 cos (θ) - 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

\frac{2 cos ( θ )}{2} = \frac{- 1}{2}

Simplify

\frac{2 cos ( θ )}{2} : cos (θ)

\frac{2 cos ( θ )}{2}

Cancel the common factor:

2

= cos (θ)

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

cos (θ) = - \frac{2}{2}

cos (θ) = - \frac{2}{2}

cos (θ) = - \frac{2}{2}

General solutions for

cos (θ) = - \frac{2}{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{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

Combine all the solutions

θ = 2 πn, θ = π + 2 πn, θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

Popular Examples

2sin(x)=2cos(2x)

2 sin (x) = 2 cos (2 x)

4sin(x/2)+2cos(x)=3

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

cos(x)= 1/5

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

sin(t)=(sqrt(3))/2

sin (t) = \frac{3}{2}

sin(θ)= 15/17

sin (θ) = \frac{15}{17}

Frequently Asked Questions (FAQ)

What is the general solution for tan(θ)=-sqrt(2)sin(θ) ?
The general solution for tan(θ)=-sqrt(2)sin(θ) is θ=2pin,θ=pi+2pin,θ=(3pi)/4+2pin,θ=(5pi)/4+2pin