Tan(t)=1/6 Q3. Find trig functions using pythag theorems 

To solve this problem, we'll use the information given to find the trigonometric functions of \( t \), specifically focusing on the third quadrant (Q3). We will also apply the Pythagorean theorem to help find the unknown sides.

### Step 1: Identify the information
You're given:
- \( \tan(t) = \frac{1}{6} \)
- The angle \( t \) lies in the third quadrant (Q3).

In Q3:
- \( \tan(t) \) is positive.
- \( \sin(t) \) and \( \cos(t) \) are both negative.

### Step 2: Use the identity for \( \tan(t) \)
We know that \( \tan(t) = \frac{\text{opposite}}{\text{adjacent}} \), so we can interpret the ratio as follows:
- Opposite side = 1
- Adjacent side = 6

Since we are in Q3, both the opposite and adjacent sides must be negative:
- Opposite side = -1
- Adjacent side = -6

### Step 3: Apply the Pythagorean theorem
We now use the Pythagorean theorem to find the hypotenuse \( r \):
\[
r = \sqrt{(-1)^2 + (-6)^2} = \sqrt{1 + 36} = \sqrt{37}
\]

### Step 4: Calculate the trig functions

1. **Sine function \( \sin(t) \):**
   \[
   \sin(t) = \frac{\text{opposite}}{r} = \frac{-1}{\sqrt{37}}
   \]

2. **Cosine function \( \cos(t) \):**
   \[
   \cos(t) = \frac{\text{adjacent}}{r} = \frac{-6}{\sqrt{37}}
   \]

3. **Tangent function \( \tan(t) \):**
   \[
   \tan(t) = \frac{\text{opposite}}{\text{adjacent}} = \frac{-1}{-6} = \frac{1}{6}
   \quad \text{(as given)}
   \]

4. **Cosecant function \( \csc(t) \):**
   \[
   \csc(t) = \frac{1}{\sin(t)} = \frac{\sqrt{37}}{-1} = -\sqrt{37}
   \]

5. **Secant function \( \sec(t) \):**
   \[
   \sec(t) = \frac{1}{\cos(t)} = \frac{\sqrt{37}}{-6} = -\frac{\sqrt{37}}{6}
   \]

6. **Cotangent function \( \cot(t) \):**
   \[
   \cot(t) = \frac{1}{\tan(t)} = 6
   \]

### Summary of results:
- \( \sin(t) = \frac{-1}{\sqrt{37}} \)
- \( \cos(t) = \frac{-6}{\sqrt{37}} \)
- \( \tan(t) = \frac{1}{6} \)
- \( \csc(t) = -\sqrt{37} \)
- \( \sec(t) = \frac{-\sqrt{37}}{6} \)
- \( \cot(t) = 6 \)

Would you like further clarification or additional details?

### Related Questions:
1. How does the sign of trigonometric functions change in each quadrant?
2. What are the Pythagorean identities for trigonometric functions?
3. How can we find the angle \( t \) when given \( \tan(t) \)?
4. How do reciprocal identities relate to the basic trigonometric functions?
5. How do we derive exact values of trigonometric functions for special angles?

**Tip:** Always remember that the signs of trigonometric functions depend on the quadrant in which the angle lies.

Tan(t)=1/6 Q3. Find trig functions using pythag theorems

Learn how to find the trigonometric functions in the third quadrant (Q3) for tan(t) = 1/6 using the Pythagorean theorem. This detailed solution covers sin(t), cos(t), tan(t), csc(t), sec(t), and cot(t) with step-by-step explanations. Suitable for students in Grades 10-12.

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