if tan t = 12/5 with angle t being in between 0 and 180, find sin t, cos t, sec t, csc t, and cot t

We are given \( \tan t = \frac{12}{5} \), and the angle \( t \) is between \( 0^\circ \) and \( 180^\circ \). We are asked to find \( \sin t \), \( \cos t \), \( \sec t \), \( \csc t \), and \( \cot t \).

### Step 1: Use the identity for tangent
We know that:
\[
\tan t = \frac{\sin t}{\cos t}
\]
From the given information, \( \tan t = \frac{12}{5} \), so:
\[
\frac{\sin t}{\cos t} = \frac{12}{5}
\]

### Step 2: Use the Pythagorean identity
We can use the identity:
\[
\sin^2 t + \cos^2 t = 1
\]
Let \( \sin t = 12k \) and \( \cos t = 5k \) for some constant \( k \) (since \( \frac{\sin t}{\cos t} = \frac{12}{5} \)).

Substitute these into the Pythagorean identity:
\[
(12k)^2 + (5k)^2 = 1
\]
\[
144k^2 + 25k^2 = 1
\]
\[
169k^2 = 1
\]
\[
k^2 = \frac{1}{169}, \quad k = \frac{1}{13}
\]

### Step 3: Find \( \sin t \) and \( \cos t \)
Now, substitute \( k = \frac{1}{13} \) back:
\[
\sin t = 12k = \frac{12}{13}, \quad \cos t = 5k = \frac{5}{13}
\]

Since \( t \) is between \( 0^\circ \) and \( 180^\circ \), and \( \tan t \) is positive, this implies that \( t \) is in the first quadrant (where both sine and cosine are positive).

### Step 4: Find the remaining values
Now that we have \( \sin t = \frac{12}{13} \) and \( \cos t = \frac{5}{13} \), we can find the other trigonometric values.

1. **Cotangent**:
\[
\cot t = \frac{1}{\tan t} = \frac{5}{12}
\]

2. **Secant**:
\[
\sec t = \frac{1}{\cos t} = \frac{13}{5}
\]

3. **Cosecant**:
\[
\csc t = \frac{1}{\sin t} = \frac{13}{12}
\]

### Summary of the Results:
- \( \sin t = \frac{12}{13} \)
- \( \cos t = \frac{5}{13} \)
- \( \tan t = \frac{12}{5} \) (given)
- \( \cot t = \frac{5}{12} \)
- \( \sec t = \frac{13}{5} \)
- \( \csc t = \frac{13}{12} \)

Would you like more details on any step or have further questions?

Here are 5 related questions:
1. How can you derive the values of secant and cosecant from sine and cosine?
2. What does the Pythagorean identity tell us about the relationship between sine and cosine?
3. How do you determine in which quadrant the angle lies based on tangent's value?
4. Can you explain the significance of inverse trigonometric functions in solving similar problems?
5. What are some real-world applications of trigonometric functions like sine and cosine?

**Tip**: Knowing which quadrant an angle lies in can help determine the sign of each trigonometric function.

If tan t = 12/5 with angle t being in between 0 and 180, find sin t, cos t, sec t, csc t, and cot t.

Solve a trigonometry problem involving tan t = 12/5 to find sin t, cos t, sec t, csc t, and cot t. Learn how to use the Pythagorean identity and quadrant analysis to derive these values. Suitable for high school students in Grades 9-12.

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