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

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

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Popular >

2tan(θ)=sin(θ)

Solution

2 tan (θ) = sin (θ)

Solution

θ = 2 πn, θ = π + 2 πn

Degrees

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

Solution steps

2 tan (θ) = sin (θ)

Subtract

sin (θ)

from both sides

2 tan (θ) - sin (θ) = 0

Express with sin, cos

- sin (θ) + 2 tan (θ)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply

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

2 \cdot \frac{sin ( θ )}{cos ( θ )}

Multiply fractions:

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

= \frac{sin ( θ ) \cdot 2}{cos ( θ )}

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

Convert element to fraction:

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

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

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

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

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

2 sin (θ) - cos (θ) sin (θ) = 0

Factor

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

2 sin (θ) - cos (θ) sin (θ)

Factor out common term

- sin (θ)

= - sin (θ) (- 2 + cos (θ))

- sin (θ) (cos (θ) - 2) = 0

Solving each part separately

sin (θ) = 0 or cos (θ) - 2 = 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

cos (θ) - 2 = 0 :

No Solution

cos (θ) - 2 = 0

Move

2

to the right side

cos (θ) - 2 = 0

Add

2

to both sides

cos (θ) - 2 + 2 = 0 + 2

Simplify

cos (θ) = 2

cos (θ) = 2

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

θ = 2 πn, θ = π + 2 πn

Popular Examples

csc(θ)=-1

csc (θ) = - 1

tan(θ)=-1/3

tan (θ) = - \frac{1}{3}

3cos^2(x)-2=0

3 cos^{2} (x) - 2 = 0

sin(x)=0.9

sin (x) = 0.9

1/2 =sin(x)

\frac{1}{2} = sin (x)

Frequently Asked Questions (FAQ)

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