Question

Solve for θ
1.5 = (sin(θ/2 + 60/2)/(sin(θ/2)

Answer 1

Assuming
`\theta` is measured in degrees, let's first convert to radians.


`\frac{60}2^\circ=30^\circ*\frac{\pi\text{ rad}}{180^\circ}=\frac\pi6`

So the equation is


`\frac32=(\sin\left(\frac\theta2+\frac\pi6\right))/(\sin\frac\theta2)`

`3\sin\frac\theta2=2\sin\left(\frac\theta2+\frac\pi6\right)`

`3\sin\frac\theta2=2\left(\sin\frac\theta2\cos\frac\pi6+\cos\frac\theta2\sin\frac\pi6\right)`

`3\sin\frac\theta2=\sqrt3\sin\frac\theta2+\cos\frac\theta2`

`3=\sqrt3+\cot\frac\theta2`

`3-\sqrt3=\cot\frac\theta2`

`\mathrm{arccot}(3-\sqrt3)+2n\pi=\frac\theta2`

`\theta=2\mathrm{arccot}(3-\sqrt3)+2n\pi`

where
`n` is any integer.