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.