if tan t = 12/5, and t is in quadrant II, find sin t, cos t, sec t, csc t, cot t

Given that \(\tan t = \frac{12}{5}\) and \(t\) is in the second quadrant (Quadrant II), we can find \(\sin t\), \(\cos t\), \(\sec t\), \(\csc t\), and \(\cot t\) using trigonometric identities.

### Step 1: Use the Pythagorean Identity

First, recall that:

\[
\tan t = \frac{\sin t}{\cos t}
\]

Since \(\tan t = \frac{12}{5}\), this means \(\sin t = 12\) and \(\cos t = 5\), but we need the correct signs and magnitudes based on \(t\) being in the second quadrant.

In the second quadrant:
- \(\sin t > 0\)
- \(\cos t < 0\)

We can treat \(\sin t = 12k\) and \(\cos t = 5k\) and find \(k\) using the Pythagorean identity:

\[
\sin^2 t + \cos^2 t = 1
\]

Substituting:

\[
(12k)^2 + (5k)^2 = 1
\]

\[
144k^2 + 25k^2 = 1
\]

\[
169k^2 = 1
\]

\[
k^2 = \frac{1}{169} \implies k = \frac{1}{13}
\]

Thus:
\[
\sin t = \frac{12}{13}, \quad \cos t = \frac{-5}{13} \quad \text{(since \(\cos t\) is negative in Quadrant II)}
\]

### Step 2: Find the other Trigonometric Functions

- \(\sin t = \frac{12}{13}\)
- \(\cos t = \frac{-5}{13}\)
- \(\tan t = \frac{12}{5}\) (already given)

Now use the reciprocal identities to find the other functions:

- \(\sec t = \frac{1}{\cos t} = \frac{1}{-\frac{5}{13}} = -\frac{13}{5}\)
- \(\csc t = \frac{1}{\sin t} = \frac{1}{\frac{12}{13}} = \frac{13}{12}\)
- \(\cot t = \frac{1}{\tan t} = \frac{1}{\frac{12}{5}} = \frac{5}{12}\)

### Final Answers:
- \(\sin t = \frac{12}{13}\)
- \(\cos t = \frac{-5}{13}\)
- \(\sec t = -\frac{13}{5}\)
- \(\csc t = \frac{13}{12}\)
- \(\cot t = \frac{5}{12}\)

Would you like more details on any of these steps, or do you have any other questions?

Here are 5 related questions:
1. How does the sign of trigonometric functions vary by quadrant?
2. What are the reciprocal identities of the six trigonometric functions?
3. Can you derive the Pythagorean identity for \(\sin^2 t + \cos^2 t = 1\)?
4. What is the significance of reference angles in different quadrants?
5. How can you solve trigonometric problems without a calculator?

**Tip:** Remember that the signs of trigonometric functions depend on the quadrant of the angle!

Learn how to find sin t, cos t, sec t, csc t, and cot t given tan t = 12/5 and that the angle is in Quadrant II. This solution involves using the Pythagorean identity and reciprocal trigonometric functions to determine each value. Perfect for high school students in Grades 9-12 studying trigonometry.

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