What is the general solution for solvefor θ,r=6sin(2θ) ?

The general solution for solvefor θ,r=6sin(2θ) is θ=(arcsin(r/6))/2+pin,θ= pi/2+(arcsin(-r/6))/2+pin

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solvefor θ,r=6sin(2θ)

solve for

θ, r = 6 sin (2 θ)

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

Solution steps

r = 6 sin (2 θ)

Switch sides

6 sin (2 θ) = r

Divide both sides by

6

6 sin (2 θ) = r

Divide both sides by

6

\frac{6 sin ( 2 θ )}{6} = \frac{r}{6}

Simplify

sin (2 θ) = \frac{r}{6}

sin (2 θ) = \frac{r}{6}

Apply trig inverse properties

sin (2 θ) = \frac{r}{6}

General solutions for

sin (2 θ) = \frac{r}{6}

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

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

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

Solve

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

2 θ = arcsin (\frac{r}{6}) + 2 πn

Divide both sides by

2

2 θ = arcsin (\frac{r}{6}) + 2 πn

Divide both sides by

2

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

Simplify

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

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

Solve

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

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

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

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

1.5 sin (3 0^{\circ}) = sin (x)

3 sin (x) - 3 cos (x) = 6

tan (θ) = \frac{11}{100}

sin (x) = \frac{4}{41}

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

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