Given:

• $\tan(\theta) = \frac{6}{5}$
• $\sin(\theta) < 0$

We need to find $\sin(\theta)$, $\cos(\theta)$, and $\sec(\theta)$.

Step-by-Step Solution

1. Determine the Quadrant:
   Since $\tan(\theta) > 0$ and $\sin(\theta) < 0$, $\theta$ must be in the third quadrant, where both sine is negative and tangent is positive.

2. Set up the Triangle:
   $\tan(\theta) = \frac{6}{5}$ can be interpreted as the ratio of the opposite side to the adjacent side in a right triangle:

   • Opposite side = $6$
   • Adjacent side = $5$

3. Calculate the Hypotenuse: Using the Pythagorean theorem: $\text{Hypotenuse} = \sqrt{6^2 + 5^2} = \sqrt{36 + 25} = \sqrt{61}$

4. Find $\sin(\theta)$, $\cos(\theta)$, and $\sec(\theta)$: Since $\theta$ is in the third quadrant:

   • $\sin(\theta) = -\frac{\text{opposite}}{\text{hypotenuse}} = -\frac{6}{\sqrt{61}} \approx -0.8$ (This matches the correct answer given.)
   • $\cos(\theta) = -\frac{\text{adjacent}}{\text{hypotenuse}} = -\frac{5}{\sqrt{61}} \approx -0.6$
   • $\sec(\theta) = \frac{1}{\cos(\theta)} \approx \frac{1}{-0.6} \approx -1.6$

Final Answers (Rounded to 1 Decimal Place)

• (a) $\sin(\theta) = -0.8$ (Correct)
• (b) $\cos(\theta) = -0.6$ (Correction needed, answer given was $0.7$)
• (c) $\sec(\theta) = -1.6$ (Correction needed, answer given was $1.6$)

Would you like further clarification or have additional questions?

Related Questions

1. How would you determine the quadrant of an angle given its trigonometric signs?
2. What is the significance of the Pythagorean identity in verifying trigonometric functions?
3. How does knowing the tangent help in finding sine and cosine values?
4. What other angles could have the same sine and cosine values but different signs?
5. How can secant be used to verify cosine values in trigonometric problems?

Tip

Always check the quadrant of the angle to determine the correct signs for each trigonometric function.