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### Question 1:
**In \( \triangle ABC \), right angle at \( B \), \( AB = 12 \text{ cm} \), and \( BC = 5 \text{ cm} \). Find the values of \( \sin A \) and \( \cos A \) and \( \tan A \) and \( \sin C \), \( \cos C \), and \( \tan C \).**

Given:
- \( AB = 12 \text{ cm} \)
- \( BC = 5 \text{ cm} \)
- \( \angle B = 90^\circ \)

Using the Pythagorean theorem:

\[
AC = \sqrt{AB^2 + BC^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \text{ cm}
\]

Now, calculate the trigonometric ratios:

For \( \angle A \):

\[
\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{5}{13}
\]
\[
\cos A = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AB}{AC} = \frac{12}{13}
\]
\[
\tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AB} = \frac{5}{12}
\]

For \( \angle C \):

\[
\sin C = \frac{AB}{AC} = \frac{12}{13}
\]
\[
\cos C = \frac{BC}{AC} = \frac{5}{13}
\]
\[
\tan C = \frac{AB}{BC} = \frac{12}{5}
\]

### Question 2:
**Given \( \sin \theta = \frac{3}{5} \), find all other trigonometric ratios.**

Since \( \sin \theta = \frac{3}{5} \), let's assume a right triangle where:
- Opposite side = 3
- Hypotenuse = 5

To find the adjacent side, use the Pythagorean theorem:

\[
\text{Adjacent side} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4
\]

Now, calculate the other trigonometric ratios:

\[
\cos \theta = \frac{4}{5}, \quad \tan \theta = \frac{3}{4}
\]
\[
\csc \theta = \frac{1}{\sin \theta} = \frac{5}{3}, \quad \sec \theta = \frac{1}{\cos \theta} = \frac{5}{4}, \quad \cot \theta = \frac{1}{\tan \theta} = \frac{4}{3}
\]

### Question 3:
**If \( \sin A = \frac{4}{5} \), evaluate \( \frac{1 - \sin^2 A}{\cos^2 A} \).**

Given \( \sin A = \frac{4}{5} \), we have:

\[
\sin^2 A = \left(\frac{4}{5}\right)^2 = \frac{16}{25}
\]

So,

\[
1 - \sin^2 A = 1 - \frac{16}{25} = \frac{9}{25}
\]

\( \cos^2 A = 

Explore detailed solutions to trigonometry problems involving right triangles. This includes finding trigonometric ratios such as sine, cosine, and tangent based on given sides and angles. Suitable for high school students studying trigonometry.

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