Find the following trigonometric ratios of \( \theta \) if \( \begin{aligned} \sin (\theta) & =\frac{4}{5} \text { and } \cos (\theta)=\frac{3}{5} \text {. Answer exactly. } \\ \tan (\theta) & =\square \\ \cot (\theta) & =\square \\ \sec (\theta) & =\square \\ \csc (\theta) & =\square\end{aligned} \)

Given: \[ \sin(\theta) = \frac{4}{5} \quad \text{and} \quad \cos(\theta) = \frac{3}{5} \] We can find the remaining trigonometric ratios using these values. 1. **Tangent (\(\tan(\theta)\))**: \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \] 2. **Cotangent (\(\cot(\theta)\))**: \[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{1}{\frac{4}{3}} = \frac{3}{4} \] 3. **Secant (\(\sec(\theta)\))**: \[ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{\frac{3}{5}} = \frac{5}{3} \] 4. **Cosecant (\(\csc(\theta)\))**: \[ \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{\frac{4}{5}} = \frac{5}{4} \] **Summary of the trigonometric ratios:** \[ \begin{aligned} \tan(\theta) &= \frac{4}{3} \\ \cot(\theta) &= \frac{3}{4} \\ \sec(\theta) &= \frac{5}{3} \\ \csc(\theta) &= \frac{5}{4} \end{aligned} \]