What is the general solution for sin(2θ)=(-2sqrt(6))/7 ?

The general solution for sin(2θ)=(-2sqrt(6))/7 is θ=-(0.77519…)/2+pin,θ= pi/2+(0.77519…)/2+pin

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sin(2θ)=(-2sqrt(6))/7

sin (2 θ) = \frac{- 2 6}{7}

θ = - \frac{0.77519 \dots}{2} + πn, θ = \frac{π}{2} + \frac{0.77519 \dots}{2} + πn

Degrees

θ = - 22.20765 \dots^{\circ} + 18 0^{\circ} n, θ = 112.20765 \dots^{\circ} + 18 0^{\circ} n

Solution steps

sin (2 θ) = \frac{- 2 6}{7}

Apply the fraction rule:

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

sin (2 θ) = - \frac{2 6}{7}

Apply trig inverse properties

sin (2 θ) = - \frac{2 6}{7}

General solutions for

sin (2 θ) = - \frac{2 6}{7}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

2 θ = arcsin (- \frac{2 6}{7}) + 2 πn, 2 θ = π + arcsin (\frac{2 6}{7}) + 2 πn

2 θ = arcsin (- \frac{2 6}{7}) + 2 πn, 2 θ = π + arcsin (\frac{2 6}{7}) + 2 πn

Solve

2 θ = arcsin (- \frac{2 6}{7}) + 2 πn : θ = - \frac{arcsin ( \frac{2 6}{7} )}{2} + πn

2 θ = arcsin (- \frac{2 6}{7}) + 2 πn

Simplify

arcsin (- \frac{2 6}{7}) + 2 πn : - arcsin (\frac{2 6}{7}) + 2 πn

arcsin (- \frac{2 6}{7}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2 6}{7}) = - arcsin (\frac{2 6}{7})

= - arcsin (\frac{2 6}{7}) + 2 πn

2 θ = - arcsin (\frac{2 6}{7}) + 2 πn

Divide both sides by

2

2 θ = - arcsin (\frac{2 6}{7}) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = - \frac{arcsin ( \frac{2 6}{7} )}{2} + \frac{2 πn}{2}

Simplify

θ = - \frac{arcsin ( \frac{2 6}{7} )}{2} + πn

θ = - \frac{arcsin ( \frac{2 6}{7} )}{2} + πn

Solve

2 θ = π + arcsin (\frac{2 6}{7}) + 2 πn : θ = \frac{π}{2} + \frac{arcsin ( \frac{2 6}{7} )}{2} + πn

2 θ = π + arcsin (\frac{2 6}{7}) + 2 πn

Divide both sides by

2

2 θ = π + arcsin (\frac{2 6}{7}) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{π}{2} + \frac{arcsin ( \frac{2 6}{7} )}{2} + \frac{2 πn}{2}

Simplify

θ = \frac{π}{2} + \frac{arcsin ( \frac{2 6}{7} )}{2} + πn

θ = \frac{π}{2} + \frac{arcsin ( \frac{2 6}{7} )}{2} + πn

θ = - \frac{arcsin ( \frac{2 6}{7} )}{2} + πn, θ = \frac{π}{2} + \frac{arcsin ( \frac{2 6}{7} )}{2} + πn

Show solutions in decimal form

θ = - \frac{0.77519 \dots}{2} + πn, θ = \frac{π}{2} + \frac{0.77519 \dots}{2} + πn

x, sin (x) = - \frac{7}{25}

\frac{1}{tan ( α )} + tan (α) = \frac{1}{sin ( α )}

tan (2 θ) = \frac{- 81}{81}

sin (θ) cos (2 θ) - cos (2 θ) = 0

tan (θ) = - \frac{7}{6}

What is the general solution for sin(2θ)=(-2sqrt(6))/7 ?
The general solution for sin(2θ)=(-2sqrt(6))/7 is θ=-(0.77519…)/2+pin,θ= pi/2+(0.77519…)/2+pin