R: Find the exact value of each of the remaining trigonometric functions of $$ heta $$. Rationalize denominators when applicable.

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R: Find the exact value of each of the remaining trigonometric functions of $$ 	heta $$. Rationalize denominators when applicable.

UpStudy AI · Accepted Answer

It looks like you want to find the exact values of all trigonometric functions for a given angle $$ \theta $$. However, to provide a precise answer, I need some additional information about $$ \theta $$. Typically, this involves knowing one or more trigonometric function values (like $$ \sin \theta $$, $$ \cos \theta $$, or $$ \tan \theta $$) or specific information about the angle itself (such as its measure or its position on the unit circle).

**Here’s how you can proceed once you have the necessary information:**

1. **Given One Trigonometric Function:**
   - **Sine ($$ \sin \theta $$) or Cosine ($$ \cos \theta $$):**
     - Use the Pythagorean identity: $$ \sin^2 \theta + \cos^2 \theta = 1 $$ to find the other.
     - Determine the remaining functions using the basic definitions:
       - $$ \tan \theta = \frac{\sin \theta}{\cos \theta} $$
       - $$ \cot \theta = \frac{\cos \theta}{\sin \theta} $$
       - $$ \sec \theta = \frac{1}{\cos \theta} $$
       - $$ \csc \theta = \frac{1}{\sin \theta} $$

2. **Given an Angle’s Position:**
   - Identify the quadrant in which $$ \theta $$ lies.
   - Determine the signs of each trigonometric function based on the quadrant.
   - Use reference angles to find exact values, especially for special angles like $$ 30^\circ $$, $$ 45^\circ $$, $$ 60^\circ $$, etc.

3. **Rationalizing Denominators:**
   - When expressing trigonometric functions like $$ \sec \theta $$ or $$ \csc \theta $$, ensure the denominators are rationalized. For example:
     - If $$ \cos \theta = \frac{1}{\sqrt{2}} $$, then $$ \sec \theta = \sqrt{2} $$.

**Example Problem:**

*Find all trigonometric functions for $$ \theta $$ given that $$ \cos \theta = \frac{3}{5} $$ and $$ \theta $$ is in the first quadrant.*

**Solution:**

1. **Given:** $$ \cos \theta = \frac{3}{5} $$.

2. **Find $$ \sin \theta $$:**
   - Use $$ \sin^2 \theta + \cos^2 \theta = 1 $$:
     $$
     \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25}
     $$
     $$
     \sin \theta = \frac{4}{5} \quad (\text{since } \theta \text{ is in the first quadrant})
     $$

3. **Find $$ \tan \theta $$:**
   $$
   \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3}
   $$

4. **Find $$ \cot \theta $$:**
   $$
   \cot \theta = \frac{1}{\tan \theta} = \frac{3}{4}
   $$

5. **Find $$ \sec \theta $$:**
   $$
   \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{3}{5}} = \frac{5}{3}
   $$

6. **Find $$ \csc \theta $$:**
   $$
   \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{4}{5}} = \frac{5}{4}
   $$

**Summary of All Trigonometric Functions:**
$$
\sin \theta = \frac{4}{5}, \quad \cos \theta = \frac{3}{5}, \quad \tan \theta = \frac{4}{3}, \quad \cot \theta = \frac{3}{4}, \quad \sec \theta = \frac{5}{3}, \quad \csc \theta = \frac{5}{4}
$$

*All denominators are rationalized as required.*

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**If you provide more specific information about $$ \theta $$, I can offer a more tailored solution!**
To find all trigonometric functions of $$ \theta $$, you need to know at least one trigonometric value of $$ \theta $$ and its quadrant. Once you have that, use the Pythagorean identity and basic definitions to determine the remaining functions.

R: Find the exact value of each of the remaining trigonometric functions of \( \theta \). Rationalize denominators when applicable.

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