What is the general solution for -0.6=sin(2θ) ?

The general solution for -0.6=sin(2θ) is θ=-(0.64350…)/2+pin,θ= pi/2+(0.64350…)/2+pin

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-0.6=sin(2θ)

- 0.6 = sin (2 θ)

θ = - \frac{0.64350 \dots}{2} + πn, θ = \frac{π}{2} + \frac{0.64350 \dots}{2} + πn

Degrees

θ = - 18.43494 \dots^{\circ} + 18 0^{\circ} n, θ = 108.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

- 0.6 = sin (2 θ)

Switch sides

sin (2 θ) = - 0.6

Apply trig inverse properties

sin (2 θ) = - 0.6

General solutions for

sin (2 θ) = - 0.6

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

2 θ = arcsin (- 0.6) + 2 πn, 2 θ = π + arcsin (0.6) + 2 πn

2 θ = arcsin (- 0.6) + 2 πn, 2 θ = π + arcsin (0.6) + 2 πn

Solve

2 θ = arcsin (- 0.6) + 2 πn : θ = - \frac{arcsin ( \frac{3}{5} )}{2} + πn

2 θ = arcsin (- 0.6) + 2 πn

Simplify

arcsin (- 0.6) + 2 πn : - arcsin (\frac{3}{5}) + 2 πn

arcsin (- 0.6) + 2 πn

arcsin (- 0.6) = - arcsin (\frac{3}{5})

arcsin (- 0.6)

= arcsin (- \frac{3}{5})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{3}{5}) = - arcsin (\frac{3}{5})

= - arcsin (\frac{3}{5})

= - arcsin (\frac{3}{5}) + 2 πn

2 θ = - arcsin (\frac{3}{5}) + 2 πn

Divide both sides by

2

2 θ = - arcsin (\frac{3}{5}) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = - \frac{arcsin ( \frac{3}{5} )}{2} + \frac{2 πn}{2}

Simplify

θ = - \frac{arcsin ( \frac{3}{5} )}{2} + πn

θ = - \frac{arcsin ( \frac{3}{5} )}{2} + πn

Solve

2 θ = π + arcsin (0.6) + 2 πn : θ = \frac{π}{2} + \frac{arcsin ( 0.6 )}{2} + πn

2 θ = π + arcsin (0.6) + 2 πn

Divide both sides by

2

2 θ = π + arcsin (0.6) + 2 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{π}{2} + \frac{arcsin ( 0.6 )}{2} + \frac{2 πn}{2}

Simplify

θ = \frac{π}{2} + \frac{arcsin ( 0.6 )}{2} + πn

θ = \frac{π}{2} + \frac{arcsin ( 0.6 )}{2} + πn

θ = - \frac{arcsin ( \frac{3}{5} )}{2} + πn, θ = \frac{π}{2} + \frac{arcsin ( 0.6 )}{2} + πn

Show solutions in decimal form

θ = - \frac{0.64350 \dots}{2} + πn, θ = \frac{π}{2} + \frac{0.64350 \dots}{2} + πn

What is the general solution for -0.6=sin(2θ) ?
The general solution for -0.6=sin(2θ) is θ=-(0.64350…)/2+pin,θ= pi/2+(0.64350…)/2+pin