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

Question

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

UpStudy AI · Accepted Answer

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:**
$$
\begin{aligned}
\tan(\theta) &= \frac{4}{3} \
\cot(\theta) &= \frac{3}{4} \
\sec(\theta) &= \frac{5}{3} \
\csc(\theta) &= \frac{5}{4}
\end{aligned}
$$
$$ \begin{aligned} \tan(\theta) &= \frac{4}{3} \ \cot(\theta) &= \frac{3}{4} \ \sec(\theta) &= \frac{5}{3} \ \csc(\theta) &= \frac{5}{4} \end{aligned} $$

Find the following trigonometric ratios of if

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