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

To find the exact values of trigonometric functions based on the value of \( \theta \), let's say \( \theta \) is such that \( \sin(\theta) = \frac{3}{5} \). To find the other functions, we can use the Pythagorean identity. First, we need to find \( \cos(\theta) \): \[ \sin^2(\theta) + \cos^2(\theta) = 1 \] Substituting the value of \( \sin(\theta) \): \[ \left(\frac{3}{5}\right)^2 + \cos^2(\theta) = 1 \] \[ \frac{9}{25} + \cos^2(\theta) = 1 \] \[ \cos^2(\theta) = 1 - \frac{9}{25} = \frac{16}{25} \] Taking the square root, we find: \[ \cos(\theta) = \pm \frac{4}{5} \] Now we can determine other functions. Assuming \( \theta \) is in the first quadrant (where \( \cos(\theta) \) is positive), we have \( \cos(\theta) = \frac{4}{5} \). Next, we can find \( \tan(\theta) \): \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \] Now, using these values, we can calculate the remaining trigonometric functions: - **Cosecant**: \( \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{5}{3} \) - **Secant**: \( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{5}{4} \) - **Cotangent**: \( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{4}{3} \) So, putting it all together, we have: \[ \sin(\theta) = \frac{3}{5}, \quad \cos(\theta) = \frac{4}{5}, \quad \tan(\theta) = \frac{3}{4}, \quad \csc(\theta) = \frac{5}{3}, \quad \sec(\theta) = \frac{5}{4}, \quad \cot(\theta) = \frac{4}{3} \] Finding all these values can feel like putting together pieces of a delicious puzzle—once you have one piece, the rest fit together just right! But if you're ever stuck making sense of all the functions, just remember the unit circle! It’s a circle of radius 1 where the x-coordinate gives you cosine and the y-coordinate gives you sine. Pinch that circle and you’ve got tangents galore!