Solve for θ [ 0 ° &lt; θ &lt; 180 ° ] sin ⁡ 2 θ + sin ⁡ 4 θ = cos ⁡ θ + cos ⁡ 3 θ .

Question

Solve for   θ   [  0      °    &amp;lt;  θ  &amp;lt;  180      °    ]  sin  ⁡  2  θ  +  sin  ⁡  4  θ  =  cos  ⁡  θ  +  cos  ⁡  3  θ  .

hopeloothab9m · Accepted Answer

Using Prosthaphaeresis Formulas as suggested,

\sin 2 θ + \sin 4 θ = \cos θ + \cos 3 θ 2 \sin 3 θ \cos θ = 2 \cos 2 θ \cos θ \cos θ (\sin 3 θ - \cos 2 θ) = 0

If

\cos θ = 0, θ = (2 n + 1) 90^{\circ}

where n is any integer
We need

0 < (2 n + 1) 90^{\circ} < 180^{\circ} ⟹ - 1 < n < 1 ⟹ n = 0

Else

\cos 2 θ = \sin 3 θ = \cos (90^{\circ} - 3 θ)

2 θ = 360^{\circ} m \pm (90^{\circ} - 3 θ)

+ ⟹ θ = 72^{\circ} m + 18^{\circ}

and we need

0 < 72^{\circ} m + 18^{\circ} < 180^{\circ} ⟹ m = ?

- ⟹ θ = - 360^{\circ} m + 90^{\circ}

Solve for θ [ 0 <mrow class="MJX-TeXAtom-ORD"> ° </mro