Find $$ \cot ( heta) $$ if $$ \sin ( heta)=\frac{5}{13} $$ and $$ heta $$ terminates in QI. Answer exactly, $$ \cot ( heta)=\square $$

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Find $$ \cot (	heta) $$ if $$ \sin (	heta)=\frac{5}{13} $$ and $$ 	heta $$ terminates in QI. Answer exactly, $$ \cot (	heta)=\square $$

UpStudy AI · Accepted Answer

Given that $$\sin(\theta) = \frac{5}{13}$$ and $$\theta$$ is in the first quadrant (QI), we can find $$\cot(\theta)$$ as follows:

1. **Find $$\cos(\theta)$$:**

Using the Pythagorean identity:
   $$
   \sin^2(\theta) + \cos^2(\theta) = 1
   $$
   Substitute $$\sin(\theta) = \frac{5}{13}$$:
   $$
   \left(\frac{5}{13}\right)^2 + \cos^2(\theta) = 1
   $$
   $$
   \frac{25}{169} + \cos^2(\theta) = 1
   $$
   $$
   \cos^2(\theta) = 1 - \frac{25}{169} = \frac{144}{169}
   $$
   Since $$\theta$$ is in QI, $$\cos(\theta)$$ is positive:
   $$
   \cos(\theta) = \frac{12}{13}
   $$

2. **Compute $$\cot(\theta)$$:**
   $$
   \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{\frac{12}{13}}{\frac{5}{13}} = \frac{12}{5}
   $$

**Final Answer:**  
$$\cot (\theta) = \dfrac{12}{5}$$
$$\cot (\theta) = \dfrac{12}{5}$$

Find if and terminates in QI. Answer exactly,

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