3) \( 5.52+3 \csc \theta=5+\frac{11}{4} \cdot \csc \theta \)

To solve the equation \( 5.52 + 3 \csc \theta = 5 + \frac{11}{4} \cdot \csc \theta \), we first want to isolate the \(\csc \theta\) terms on one side. Rearranging gives: \[ 3 \csc \theta - \frac{11}{4} \csc \theta = 5 - 5.52 \] This simplifies to: \[ \left(3 - \frac{11}{4}\right) \csc \theta = -0.52 \] Finding a common denominator for the left side, we rewrite \(3\) as \(\frac{12}{4}\): \[ \left(\frac{12}{4} - \frac{11}{4}\right) \csc \theta = -0.52 \] This further simplifies to: \[ \frac{1}{4} \csc \theta = -0.52 \] Multiplying both sides by 4 results in: \[ \csc \theta = -2.08 \] The cosecant function, \(\csc \theta\), is the reciprocal of the sine function, so we have: \[ \sin \theta = -\frac{1}{2.08} \] Now, you can calculate the value of \(\sin \theta\) and find the corresponding angles based on this value. Happy calculating!