If tant=−815tant=-815 and π2π2 <t<<t< ππ, find sintsint, costcost, sectsect, csctcsct, cott

To solve for \(\sin t\), \(\cos t\), \(\sec t\), \(\csc t\), and \(\cot t\) given \(\tan t = -\frac{8}{15}\) and \(\frac{\pi}{2} < t < \pi\), follow these steps:

1. **Determine the quadrant:**
   - Since \(\frac{\pi}{2} < t < \pi\), \(t\) is in the second quadrant.
   - In the second quadrant, \(\sin t\) is positive and \(\cos t\) is negative.

2. **Use \(\tan t\) to find \(\sin t\) and \(\cos t\):**
   - \(\tan t = \frac{\sin t}{\cos t} = -\frac{8}{15}\).
   - Represent this as a right triangle where the opposite side is 8 (since \(\sin t\) is positive in the second quadrant) and the adjacent side is 15 (since \(\cos t\) is negative in the second quadrant).

3. **Calculate the hypotenuse:**
   \[
   \text{hypotenuse} = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17
   \]

4. **Find \(\sin t\) and \(\cos t\):**
   \[
   \sin t = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{8}{17}
   \]
   \[
   \cos t = \frac{\text{adjacent}}{\text{hypotenuse}} = -\frac{15}{17}
   \]

5. **Calculate the remaining trigonometric functions:**
   - \(\sec t = \frac{1}{\cos t} = \frac{1}{-\frac{15}{17}} = -\frac{17}{15}\)
   - \(\csc t = \frac{1}{\sin t} = \frac{1}{\frac{8}{17}} = \frac{17}{8}\)
   - \(\cot t = \frac{1}{\tan t} = \frac{1}{-\frac{8}{15}} = -\frac{15}{8}\)

Thus, the exact values are:
\[
\sin t = \frac{8}{17}
\]
\[
\cos t = -\frac{15}{17}
\]
\[
\sec t = -\frac{17}{15}
\]
\[
\csc t = \frac{17}{8}
\]
\[
\cot t = -\frac{15}{8}
\]

Do you want details or have any questions?

Here are 8 related questions to expand on this topic:
1. What is the value of \(\sin t\) if \(\cot t = \frac{7}{24}\) and \(\pi < t < \frac{3\pi}{2}\)?
2. How do you find \(\cos t\) given \(\tan t\) and the quadrant?
3. What are the values of \(\sec t\) and \(\csc t\) if \(\sin t = -\frac{5}{13}\) and \(t\) is in the third quadrant?
4. How do you determine the signs of trigonometric functions based on the angle's quadrant?
5. Can you find \(\cot t\) if \(\sin t\) and \(\cos t\) are known?
6. What is the relationship between \(\tan t\) and \(\cot t\)?
7. How do you use the Pythagorean identity to find \(\sin t\) given \(\cos t\)?
8. Why is it important to consider the quadrant when solving trigonometric problems?

**Tip:** Always double-check the signs of your trigonometric functions based on the quadrant in which the angle lies.

Learn how to solve for sine, cosine, secant, cosecant, and cotangent when given tan(t) = -8/15 and π/2 < t < π. Detailed step-by-step solution including trigonometric identities and quadrant analysis. Suitable for high school students studying trigonometry.

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