We are given that $\tan(\theta) = \frac{8}{15}$ and that $\theta$ is in Quadrant III. We need to find the exact values of the other trigonometric functions: $\sin(\theta)$, $\cos(\theta)$, $\sec(\theta)$, $\csc(\theta)$, and $\cot(\theta)$.

Step 1: Use the identity $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

We know: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{8}{15}$ So we have: $\frac{\sin(\theta)}{\cos(\theta)} = \frac{8}{15}$

Step 2: Use the Pythagorean identity

The Pythagorean identity states: $\sin^2(\theta) + \cos^2(\theta) = 1$

To use this identity, we need to express $\sin(\theta)$ and $\cos(\theta)$ in terms of a hypotenuse.

Step 3: Find $\sin(\theta)$ and $\cos(\theta)$ using a right triangle

Since $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$, we can construct a right triangle where the opposite side is 8 and the adjacent side is 15. Using the Pythagorean theorem to find the hypotenuse $r$: $r = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17$

Step 4: Determine the signs of $\sin(\theta)$ and $\cos(\theta)$

Since $\theta$ is in Quadrant III, both $\sin(\theta)$ and $\cos(\theta)$ are negative. Therefore: $\sin(\theta) = \frac{-8}{17}, \quad \cos(\theta) = \frac{-15}{17}$

Step 5: Calculate the remaining trigonometric functions

Now, we can find the remaining trigonometric functions:

1. $\sec(\theta)$: Reciprocal of $\cos(\theta)$ $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{-\frac{15}{17}} = -\frac{17}{15}$

2. $\csc(\theta)$: Reciprocal of $\sin(\theta)$ $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{-\frac{8}{17}} = -\frac{17}{8}$

3. $\cot(\theta)$: Reciprocal of $\tan(\theta)$ $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{1}{\frac{8}{15}} = \frac{15}{8}$

Final Results

• $\sin(\theta) = \frac{-8}{17}$
• $\cos(\theta) = \frac{-15}{17}$
• $\sec(\theta) = \frac{-17}{15}$
• $\csc(\theta) = \frac{-17}{8}$
• $\cot(\theta) = \frac{15}{8}$

Would you like any further details on these calculations?

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Related Questions:

1. What is the geometric interpretation of trigonometric functions in each quadrant?
2. How does the Pythagorean identity relate to the unit circle?
3. Why are sine and cosine both negative in Quadrant III?
4. What are the general formulas for trigonometric functions in different quadrants?
5. How does the reciprocal identity help in finding trigonometric functions?

Tip: When solving trigonometric problems in different quadrants, always remember the sign rules for each function in the four quadrants.