We are given:

$\sin \theta = \frac{4}{5}, \quad \cos \theta < 0$

Step 1: Determine the quadrant of $\theta$

Since $\sin \theta > 0$ and $\cos \theta < 0$, the angle $\theta$ lies in the second quadrant.

Step 2: Use the Pythagorean identity to find $\cos \theta$

The Pythagorean identity is: $\sin^2 \theta + \cos^2 \theta = 1$

Substitute $\sin \theta = \frac{4}{5}$: $\left( \frac{4}{5} \right)^2 + \cos^2 \theta = 1$

$\frac{16}{25} + \cos^2 \theta = 1$

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

$\cos \theta = \pm \frac{3}{5}$

Since $\cos \theta < 0$, we take: $\cos \theta = -\frac{3}{5}$

Step 3: Find the other trigonometric functions

1. Tangent ($\tan \theta$)

$\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{4}{5}}{-\frac{3}{5}} = -\frac{4}{3}$

2. Cosecant ($\csc \theta$)

$\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{4}{5}} = \frac{5}{4}$

3. Secant ($\sec \theta$)

$\sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{3}{5}} = -\frac{5}{3}$

4. Cotangent ($\cot \theta$)

$\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{-\frac{3}{5}}{\frac{4}{5}} = -\frac{3}{4}$

Step 4: Find the reference angle

The reference angle is the acute angle formed with the x-axis. Since $\sin \theta = \frac{4}{5}$, we use: $\sin^{-1} \left( \frac{4}{5} \right)$

Using approximate values: $\text{Reference angle} \approx 53.13^\circ$

Final Answer:

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

Reference angle: $53.13^\circ$.

Would you like me to break this down further? Here are some related questions for deeper understanding:

1. How can you verify the quadrant for a given trigonometric condition?
2. Why is the reference angle always positive?
3. How does the Pythagorean identity help in finding other trigonometric functions?
4. Can you apply these calculations to radians?
5. What is the geometric interpretation of the given values?

Tip: Memorize common trigonometric values for angles like $30^\circ$, $45^\circ$, and $60^\circ$ to speed up calculations.