We are tasked with finding the exact values of $\tan\theta$ and $\sec\theta$ given the conditions:

1. $\sin\theta = -\frac{3}{4}$
2. $\cot\theta > 0$.

Step 1: Analyze the quadrant

• Since $\sin\theta = -\frac{3}{4}$, $\theta$ must be in either the third or fourth quadrant, where sine is negative.
• Given $\cot\theta > 0$, and since $\cot\theta = \cos\theta / \sin\theta$, this means that $\cos\theta$ and $\sin\theta$ have the same sign. This only happens in the third quadrant (where both sine and cosine are negative).
  Thus, $\theta$ is in the third quadrant.

Step 2: Use the Pythagorean identity

The Pythagorean identity is:
$\sin^2\theta + \cos^2\theta = 1$
Substituting $\sin\theta = -\frac{3}{4}$:
$\left(-\frac{3}{4}\right)^2 + \cos^2\theta = 1$
$\frac{9}{16} + \cos^2\theta = 1$
$\cos^2\theta = 1 - \frac{9}{16} = \frac{16}{16} - \frac{9}{16} = \frac{7}{16}$
$\cos\theta = -\sqrt{\frac{7}{16}} = -\frac{\sqrt{7}}{4}$
(The cosine is negative because $\theta$ is in the third quadrant.)

Step 3: Find $\tan\theta$

$\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{-\frac{3}{4}}{-\frac{\sqrt{7}}{4}} = \frac{3}{\sqrt{7}} = \frac{3\sqrt{7}}{7}$

Step 4: Find $\sec\theta$

The secant is the reciprocal of cosine:
$\sec\theta = \frac{1}{\cos\theta} = \frac{1}{-\frac{\sqrt{7}}{4}} = -\frac{4}{\sqrt{7}} = -\frac{4\sqrt{7}}{7}$

Final Answers:

• $\tan\theta = \frac{3\sqrt{7}}{7}$
• $\sec\theta = -\frac{4\sqrt{7}}{7}$

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Would you like a step-by-step explanation of the trigonometric reasoning or more examples?

Here are some related questions:

1. What are the values of $\cot\theta$ and $\csc\theta$ for this angle?
2. How do signs of trigonometric functions change in each quadrant?
3. What is the significance of the Pythagorean identity in trigonometry?
4. How can you determine an angle’s quadrant based on trigonometric values?
5. What is the relationship between reciprocal trigonometric functions?

Tip: Always check the quadrant carefully before determining the signs of trigonometric functions!