FimDinia the alriera sonution 1. Deicrmine the valucs of $$ heta $$, for $$ -90^{\circ}

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FimDinia the alriera sonution 1. Deicrmine the valucs of $$ 	heta $$, for $$ -90^{\circ}<0<360^{\circ} $$ 1. $$ \operatorname{sir}(	heta+100)=0,757 $$

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1. Let 
   $$
   x=\theta+100.
   $$
   Then the equation becomes
   $$
   \sin(x)=0.757.
   $$

2. Compute the principal value:
   $$
   x_1=\sin^{-1}(0.757).
   $$
   Using a calculator,
   $$
   x_1\approx 49.0^{\circ}.
   $$
   
3. The sine function is positive in the first and second quadrants. Thus, the second solution in one period is
   $$
   x_2=180^{\circ}-x_1\approx180^{\circ}-49.0^{\circ}=131.0^{\circ}.
   $$

4. The general solutions for $$x$$ are:
   $$
   x=x_1+360^{\circ}\,k \quad \text{or} \quad x=x_2+360^{\circ}\,k,\quad k\in \mathbb{Z}.
   $$

5. The relation between $$x$$ and $$\theta$$ is
   $$
   \theta=x-100.
   $$
   The given domain for $$\theta$$ is:
   $$
   -90^{\circ}<\theta<360^{\circ}.
   $$
   In terms of $$x$$, this becomes:
   \[
   -90^{\circ}+100The values of $$ \theta $$ are approximately $$-51^\circ$$, $$31^\circ$$, and $$309^\circ$$.

FimDinia the alriera sonution 1. Deicrmine the valucs of \( \theta \), for \( -90^{\circ}<0<360^{\circ} \) 1. \( \operatorname{sir}(\theta+100)=0,757 \)

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