Solve the equation sin 5/2ø= 1/2 for 0⁰<ø< 180⁰

$$\phi = 30^\circ + 360^\circ n \quad \text{or} \quad \phi = 150^\circ + 360^\circ n,$$

where $n$ is any integer.

First equation:

$$\theta = \frac{2}{5} (30^\circ + 360^\circ n) = 12^\circ + 144^\circ n.$$

Second equation:

$$\theta = \frac{2}{5} (150^\circ + 360^\circ n) = 60^\circ + 144^\circ n.$$

For $\theta = 12^\circ + 144^\circ n$:

• $n=0$: $\theta = 12^\circ$
• $n=1$: $\theta = 156^\circ$
• $n=2$: $\theta = 300^\circ > 180^\circ$
• $n=-1$: $\theta = -132^\circ < 0^\circ$

For $\theta = 60^\circ + 144^\circ n$:

• $n=0$: $\theta = 60^\circ$
• $n=1$: $\theta = 204^\circ > 180^\circ$
• $n=-1$: $\theta = -84^\circ < 0^\circ$

Solutions: $\theta = 12^\circ, 60^\circ, 156^\circ$

$$12^\circ, 60^\circ, 156^\circ$$