We are given that $\sin(\theta) = \frac{3}{5}$ and that the angle $\theta$ lies in quadrant II. To find the exact values of all six trigonometric functions, let's use the given information and trigonometric identities. Here are the steps:

Step 1: Find $\cos(\theta)$ and $\tan(\theta)$

We know that in quadrant II, sine is positive, but cosine is negative. We'll use the Pythagorean identity:

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

Substitute $\sin(\theta) = \frac{3}{5}$:

$\left(\frac{3}{5}\right)^2 + \cos^2(\theta) = 1$

$\frac{9}{25} + \cos^2(\theta) = 1$

$\cos^2(\theta) = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25}$

$\cos(\theta) = \pm \frac{4}{5}$

Since $\theta$ lies in quadrant II, $\cos(\theta)$ must be negative:

$\cos(\theta) = -\frac{4}{5}$

Step 2: Find $\tan(\theta)$

Now, we can find $\tan(\theta)$ using the identity:

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

Substitute the known values:

$\tan(\theta) = \frac{\frac{3}{5}}{-\frac{4}{5}} = -\frac{3}{4}$

Step 3: Find $\csc(\theta)$, $\sec(\theta)$, and $\cot(\theta)$

Now, we use the reciprocal identities to find the other functions:

• $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{\frac{3}{5}} = \frac{5}{3}$
• $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4}$
• $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{1}{-\frac{3}{4}} = -\frac{4}{3}$

Final Answer:

The exact values of the six trigonometric functions are:

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

Would you like a more detailed explanation of any of these steps, or do you have any follow-up questions? Here are some related questions:

1. How do I use the Pythagorean identity to find trigonometric values?
2. How do the signs of trigonometric functions change in different quadrants?
3. How can I derive trigonometric identities from basic relationships?
4. What other methods can be used to find trigonometric functions if I know one?
5. How can I visualize the relationship between sine, cosine, and tangent on the unit circle?

Tip: Remember that in quadrant II, sine is positive, cosine is negative, and tangent is negative as well.