What is the general solution for sin(θ+pi/6)=2sin(θ-pi/6) ?

The general solution for sin(θ+pi/6)=2sin(θ-pi/6) is θ= pi/3+pin

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sin(θ+pi/6)=2sin(θ-pi/6)

Solution

sin (θ + \frac{π}{6}) = 2 sin (θ - \frac{π}{6})

Solution

θ = \frac{π}{3} + πn

Degrees

θ = 6 0^{\circ} + 18 0^{\circ} n

Solution steps

sin (θ + \frac{π}{6}) = 2 sin (θ - \frac{π}{6})

Rewrite using trig identities

sin (θ + \frac{π}{6}) = 2 sin (θ - \frac{π}{6})

Rewrite using trig identities

sin (θ + \frac{π}{6})

Use the Angle Sum identity:

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

= sin (θ) cos (\frac{π}{6}) + cos (θ) sin (\frac{π}{6})

Simplify

sin (θ) cos (\frac{π}{6}) + cos (θ) sin (\frac{π}{6}) : \frac{3}{2} sin (θ) + \frac{1}{2} cos (θ)

sin (θ) cos (\frac{π}{6}) + cos (θ) sin (\frac{π}{6})

Simplify

cos (\frac{π}{6}) : \frac{3}{2}

cos (\frac{π}{6})

Use the following trivial identity:

cos (\frac{π}{6}) = \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}

= \frac{3}{2}

= \frac{3}{2} sin (θ) + sin (\frac{π}{6}) cos (θ)

Simplify

sin (\frac{π}{6}) : \frac{1}{2}

sin (\frac{π}{6})

Use the following trivial identity:

sin (\frac{π}{6}) = \frac{1}{2}

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}

= \frac{1}{2}

= \frac{3}{2} sin (θ) + \frac{1}{2} cos (θ)

= \frac{3}{2} sin (θ) + \frac{1}{2} cos (θ)

Use the Angle Difference identity:

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

= sin (θ) cos (\frac{π}{6}) - cos (θ) sin (\frac{π}{6})

Simplify

sin (θ) cos (\frac{π}{6}) - cos (θ) sin (\frac{π}{6}) : \frac{3}{2} sin (θ) - \frac{1}{2} cos (θ)

sin (θ) cos (\frac{π}{6}) - cos (θ) sin (\frac{π}{6})

Simplify

cos (\frac{π}{6}) : \frac{3}{2}

cos (\frac{π}{6})

Use the following trivial identity:

cos (\frac{π}{6}) = \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}

= \frac{3}{2}

= \frac{3}{2} sin (θ) - sin (\frac{π}{6}) cos (θ)

Simplify

sin (\frac{π}{6}) : \frac{1}{2}

sin (\frac{π}{6})

Use the following trivial identity:

sin (\frac{π}{6}) = \frac{1}{2}

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}

= \frac{1}{2}

= \frac{3}{2} sin (θ) - \frac{1}{2} cos (θ)

= \frac{3}{2} sin (θ) - \frac{1}{2} cos (θ)

\frac{3}{2} sin (θ) + \frac{1}{2} cos (θ) = 2 (\frac{3}{2} sin (θ) - \frac{1}{2} cos (θ))

Simplify

2 (\frac{3}{2} sin (θ) - \frac{1}{2} cos (θ)) : 3 sin (θ) - cos (θ)

2 (\frac{3}{2} sin (θ) - \frac{1}{2} cos (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = \frac{3}{2} sin (θ), c = \frac{1}{2} cos (θ)

= 2 \cdot \frac{3}{2} sin (θ) - 2 \cdot \frac{1}{2} cos (θ)

Simplify

2 \cdot \frac{3}{2} sin (θ) - 2 \cdot \frac{1}{2} cos (θ) : 3 sin (θ) - cos (θ)

2 \cdot \frac{3}{2} sin (θ) - 2 \cdot \frac{1}{2} cos (θ)

2 \cdot \frac{3}{2} sin (θ) = 3 sin (θ)

2 \cdot \frac{3}{2} sin (θ)

Multiply fractions:

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

= \frac{2 3}{2} sin (θ)

Cancel the common factor:

2

= sin (θ) 3

2 \cdot \frac{1}{2} cos (θ) = cos (θ)

2 \cdot \frac{1}{2} cos (θ)

Multiply fractions:

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

= \frac{1 \cdot 2}{2} cos (θ)

Cancel the common factor:

2

= cos (θ) \cdot 1

Multiply:

cos (θ) \cdot 1 = cos (θ)

= cos (θ)

= 3 sin (θ) - cos (θ)

= 3 sin (θ) - cos (θ)

\frac{3}{2} sin (θ) + \frac{1}{2} cos (θ) = 3 sin (θ) - cos (θ)

\frac{3}{2} sin (θ) + \frac{1}{2} cos (θ) = 3 sin (θ) - cos (θ)

Subtract

3 sin (θ) - cos (θ)

from both sides

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

Simplify

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

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

Multiply

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

\frac{3}{2} cos (θ)

Multiply fractions:

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

= \frac{3 cos ( θ )}{2}

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

Multiply

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

\frac{3}{2} sin (θ)

Multiply fractions:

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

= \frac{3 sin ( θ )}{2}

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

Combine the fractions

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

Apply rule

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

= \frac{3 cos ( θ ) + 3 sin ( θ )}{2}

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

Convert element to fraction:

3 sin (θ) = \frac{3 sin ( θ ) 2}{2}

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

Since the denominators are equal, combine the fractions:

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

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

Add similar elements:

3 sin (θ) - 23 sin (θ) = - 3 sin (θ)

= \frac{3 cos ( θ ) - 3 sin ( θ )}{2}

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

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

3 cos (θ) - 3 sin (θ) = 0

Rewrite using trig identities

3 cos (θ) - 3 sin (θ) = 0

Divide both sides by

cos (θ), cos (θ) \neq = 0

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

Simplify

3 - \frac{3 sin ( θ )}{cos ( θ )} = 0

Use the basic trigonometric identity:

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

3 - 3 tan (θ) = 0

3 - 3 tan (θ) = 0

Move

3

to the right side

3 - 3 tan (θ) = 0

Subtract

3

from both sides

3 - 3 tan (θ) - 3 = 0 - 3

Simplify

- 3 tan (θ) = - 3

- 3 tan (θ) = - 3

Divide both sides by

- 3

- 3 tan (θ) = - 3

Divide both sides by

- 3

\frac{- 3 tan ( θ )}{- 3} = \frac{- 3}{- 3}

Simplify

\frac{- 3 tan ( θ )}{- 3} = \frac{- 3}{- 3}

Simplify

\frac{- 3 tan ( θ )}{- 3} : tan (θ)

\frac{- 3 tan ( θ )}{- 3}

Apply the fraction rule:

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

= \frac{3 tan ( θ )}{3}

Cancel the common factor:

3

= tan (θ)

Simplify

\frac{- 3}{- 3} : 3

\frac{- 3}{- 3}

Apply the fraction rule:

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

= \frac{3}{3}

Apply radical rule:

3 = 3^{\frac{1}{2}}

= \frac{3}{3 ^{\frac{1}{2}}}

Apply exponent rule:

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

\frac{3 ^{1}}{3 ^{\frac{1}{2}}} = 3^{1 - \frac{1}{2}}

= 3^{1 - \frac{1}{2}}

Subtract the numbers:

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

= 3^{\frac{1}{2}}

Apply radical rule:

3^{\frac{1}{2}} = 3

= 3

tan (θ) = 3

tan (θ) = 3

tan (θ) = 3

General solutions for

tan (θ) = 3

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

θ = \frac{π}{3} + πn

θ = \frac{π}{3} + πn

Popular Examples

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

4cos^2(θ)tan(θ)-tan(θ)=0

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1/(cos(x))-(sin(x))/(cos(x))=cos(x)

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

cos(x)=(-8)/9

cos (x) = \frac{- 8}{9}

cos(x)= 6/(7sqrt(76))

cos (x) = \frac{6}{7 76}

Frequently Asked Questions (FAQ)

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