What is the general solution for 2sin(θ)=sqrt(3)tan(θ) ?

The general solution for 2sin(θ)=sqrt(3)tan(θ) is θ=2pin,θ=pi+2pin,θ= pi/6+2pin,θ=(11pi)/6+2pin

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Popular >

2sin(θ)=sqrt(3)tan(θ)

Solution

2 sin (θ) = 3 tan (θ)

Solution

θ = 2 πn, θ = π + 2 πn, θ = \frac{π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

Degrees

θ = 0^{\circ} + 36 0^{\circ} n, θ = 18 0^{\circ} + 36 0^{\circ} n, θ = 3 0^{\circ} + 36 0^{\circ} n, θ = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sin (θ) = 3 tan (θ)

Subtract

3 tan (θ)

from both sides

2 sin (θ) - 3 tan (θ) = 0

Express with sin, cos

2 sin (θ) - 3 tan (θ)

Use the basic trigonometric identity:

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

= 2 sin (θ) - 3 \frac{sin ( θ )}{cos ( θ )}

Simplify

2 sin (θ) - 3 \frac{sin ( θ )}{cos ( θ )} : \frac{2 sin ( θ ) cos ( θ ) - 3 sin ( θ )}{cos ( θ )}

2 sin (θ) - 3 \frac{sin ( θ )}{cos ( θ )}

Multiply

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

3 \frac{sin ( θ )}{cos ( θ )}

Multiply fractions:

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

= \frac{sin ( θ ) 3}{cos ( θ )}

= 2 sin (θ) - \frac{3 sin ( θ )}{cos ( θ )}

Convert element to fraction:

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

= \frac{2 sin ( θ ) cos ( θ )}{cos ( θ )} - \frac{sin ( θ ) 3}{cos ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{2 sin ( θ ) cos ( θ ) - sin ( θ ) 3}{cos ( θ )}

= \frac{2 sin ( θ ) cos ( θ ) - 3 sin ( θ )}{cos ( θ )}

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

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

- sin (θ) 3 + 2 cos (θ) sin (θ) = 0

Factor

- sin (θ) 3 + 2 cos (θ) sin (θ) : sin (θ) (- 3 + 2 cos (θ))

- sin (θ) 3 + 2 cos (θ) sin (θ)

Factor out common term

sin (θ)

= sin (θ) (- 3 + 2 cos (θ))

sin (θ) (- 3 + 2 cos (θ)) = 0

Solving each part separately

sin (θ) = 0 or - 3 + 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

- 3 + 2 cos (θ) = 0 : θ = \frac{π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

- 3 + 2 cos (θ) = 0

Move

3

to the right side

- 3 + 2 cos (θ) = 0

Add

3

to both sides

- 3 + 2 cos (θ) + 3 = 0 + 3

Simplify

2 cos (θ) = 3

2 cos (θ) = 3

Divide both sides by

2

2 cos (θ) = 3

Divide both sides by

2

\frac{2 cos ( θ )}{2} = \frac{3}{2}

Simplify

cos (θ) = \frac{3}{2}

cos (θ) = \frac{3}{2}

General solutions for

cos (θ) = \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}

θ = \frac{π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

θ = \frac{π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

Combine all the solutions

θ = 2 πn, θ = π + 2 πn, θ = \frac{π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

Popular Examples

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sin (x^{2} - 1) = 4

1-cos(x)=(2pi)/3

1 - cos (x) = \frac{2 π}{3}

sin(A)= 1/3

sin (A) = \frac{1}{3}

sin(θ)=-0.78

sin (θ) = - 0.78

solvefor x,sin(2x)+cos(x)=0 solve for

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

Frequently Asked Questions (FAQ)

What is the general solution for 2sin(θ)=sqrt(3)tan(θ) ?
The general solution for 2sin(θ)=sqrt(3)tan(θ) is θ=2pin,θ=pi+2pin,θ= pi/6+2pin,θ=(11pi)/6+2pin