\section*{Problem 2 - V11012}

What values can \(\mathrm{x}\) take in the following equations?
a) \(\sin x=\sin 69^{\circ}\)
b) \(\tan \left(\frac{x}{2}+32^{\circ}\right)=1\)
c) \(\sin ^{2} x+\cos ^{2} \frac{x}{2}=3.2\)
}

a) \(x_{1}=69^{\circ} ; x_{2}=111^{\circ}\). If the solutions are given considering periodicity, thus with \(k=0, \pm 1, \pm 2, \ldots\)

\[
x_{1}=69^{\circ}+k \cdot 360^{\circ} \quad ; \quad x_{2}=111^{\circ}+k \cdot 360^{\circ}
\]

b) \(x_{1}=26^{\circ} ; x_{2}=206^{\circ}\). Considering periodicity

\[
x=26^{\circ}+k \cdot 180^{\circ} \quad \text { with } k=0, \pm 1, \pm 2, \ldots
\]

c) Since both \(\sin ^{2} x\) and \(\cos ^{2} x\) can at most take the value 1, their sum can never be 3.2. Therefore, there is no \(x\) that satisfies the given conditions.

Adapted from \([6]\)