Part (a): Rough Sketch

To visualize the angle $\theta$ in Quadrant II with $\csc(\theta) = \frac{3}{2}$, let's go through the steps:

1. The cosecant function is the reciprocal of the sine function. So, we have:

$\csc(\theta) = \frac{1}{\sin(\theta)}$

Since $\csc(\theta) = \frac{3}{2}$, it follows that:

$\sin(\theta) = \frac{2}{3}$

2. In Quadrant II, the sine function is positive, and the cosine function is negative.

3. Now, using the Pythagorean identity:

$\sin^2(\theta) + \cos^2(\theta) = 1$

Substitute $\sin(\theta) = \frac{2}{3}$:

$\left(\frac{2}{3}\right)^2 + \cos^2(\theta) = 1$

$\frac{4}{9} + \cos^2(\theta) = 1$

$\cos^2(\theta) = 1 - \frac{4}{9} = \frac{5}{9}$

$\cos(\theta) = -\frac{\sqrt{5}}{3}$

Since we are in Quadrant II, $\cos(\theta)$ is negative.

4. The other trigonometric functions can be derived from the sine and cosine values.

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Part (b): Find the Values of All Six Trigonometric Functions

1. Sine: $\sin(\theta) = \frac{2}{3}$

2. Cosine: $\cos(\theta) = -\frac{\sqrt{5}}{3}$

3. Tangent: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{2}{3}}{-\frac{\sqrt{5}}{3}} = -\frac{2}{\sqrt{5}} = -\frac{2\sqrt{5}}{5}$

4. Secant (reciprocal of cosine): $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{-\frac{\sqrt{5}}{3}} = -\frac{3}{\sqrt{5}} = -\frac{3\sqrt{5}}{5}$

5. Cosecant (reciprocal of sine): $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{\frac{2}{3}} = \frac{3}{2}$

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Final Trigonometric Values:

1. $\sin(\theta) = \frac{2}{3}$
2. $\cos(\theta) = -\frac{\sqrt{5}}{3}$
3. $\tan(\theta) = -\frac{2\sqrt{5}}{5}$
4. $\sec(\theta) = -\frac{3\sqrt{5}}{5}$
5. $\csc(\theta) = \frac{3}{2}$

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Questions for Further Study:

1. How do the signs of trigonometric functions change in different quadrants?
2. Can we calculate the trigonometric functions for angles in other quadrants if we know the values of sine or cosine?
3. What is the significance of using Pythagorean identities in trigonometry?
4. How do reciprocal trigonometric functions relate to each other in terms of sine, cosine, and tangent?
5. What are some methods to quickly determine the values of sine, cosine, and tangent for special angles?

Tip: Remember that the trigonometric functions are periodic, so knowing one cycle of these functions helps to find their values for other angles that differ by multiples of $2\pi$.