Find the values of the trigonometric functions of 𝜃 from the information given.
tan(𝜃) = −3,
sin(𝜃) > 0 

Given the information:

1. \(\tan(\theta) = -3\)
2. \(\sin(\theta) > 0\)

### Step 1: Determine the quadrant
- Since \(\tan(\theta) = -3\), the tangent is negative.
- Since \(\sin(\theta) > 0\), the sine is positive.

Tangent is negative and sine is positive in **Quadrant II**.

### Step 2: Use \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)
- \(\tan(\theta) = -3\) implies \(\frac{\sin(\theta)}{\cos(\theta)} = -3\).

In Quadrant II, sine is positive and cosine is negative. Let’s assume:
\[
\sin(\theta) = 3k \quad \text{and} \quad \cos(\theta) = -k
\]
Substitute into the tangent formula:
\[
\frac{3k}{-k} = -3
\]
This satisfies the equation, so the sine and cosine values are proportional to 3 and -1.

### Step 3: Calculate \(\sin(\theta)\) and \(\cos(\theta)\)
To find the exact values, we use the identity:
\[
\sin^2(\theta) + \cos^2(\theta) = 1
\]
Substitute \(\sin(\theta) = 3k\) and \(\cos(\theta) = -k\) into the equation:
\[
(3k)^2 + (-k)^2 = 1
\]
\[
9k^2 + k^2 = 1
\]
\[
10k^2 = 1
\]
\[
k^2 = \frac{1}{10}, \quad k = \frac{1}{\sqrt{10}}
\]

### Step 4: Final values of \(\sin(\theta)\) and \(\cos(\theta)\)
- \(\sin(\theta) = 3k = \frac{3}{\sqrt{10}} = \frac{3\sqrt{10}}{10}\)
- \(\cos(\theta) = -k = -\frac{1}{\sqrt{10}} = -\frac{\sqrt{10}}{10}\)

### Step 5: Calculate the remaining trigonometric functions
- \(\tan(\theta) = -3\) (given)
- \(\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{1}{-3} = -\frac{1}{3}\)
- \(\sec(\theta) = \frac{1}{\cos(\theta)} = -\sqrt{10}\)
- \(\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\sqrt{10}}{3}\)

### Final answers:
- \(\sin(\theta) = \frac{3\sqrt{10}}{10}\)
- \(\cos(\theta) = -\frac{\sqrt{10}}{10}\)
- \(\tan(\theta) = -3\)
- \(\cot(\theta) = -\frac{1}{3}\)
- \(\sec(\theta) = -\sqrt{10}\)
- \(\csc(\theta) = \frac{\sqrt{10}}{3}\)

Would you like more details or further clarifications?

#### Related questions:
1. How do you determine the quadrant based on sine and tangent signs?
2. Can you explain the identity \(\sin^2(\theta) + \cos^2(\theta) = 1\) further?
3. How is \(\sec(\theta)\) calculated from \(\cos(\theta)\)?
4. How do inverse trigonometric functions relate to this problem?
5. What are the possible ranges for trigonometric functions in different quadrants?

#### Tip:
Always use the Pythagorean identity \(\sin^2(\theta) + \cos^2(\theta) = 1\) to verify trigonometric values.

Find the values of the trigonometric functions of 𝜃 from the information given. tan(𝜃) = −3, sin(𝜃) > 0.

Explore the step-by-step solution to find the values of trigonometric functions given tan(𝜃) = -3 and sin(𝜃) > 0. This problem involves understanding trigonometric identities, quadrant determination, and the Pythagorean identity. Ideal for high school students in Grades 10-12.

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