In a right triangle, we can find trigonometric ratios using the known sides. You are given:

• Right triangle $ABC$ with $\angle C = 90^\circ$,
• $AB = 15$ (hypotenuse),
• $AC = 9$ (adjacent to angle $A$).

To find $\sin(A)$ and $\cot(A)$, we first need to calculate the length of side $BC$, which is the opposite side to $\angle A$, using the Pythagorean theorem:

$AB^2 = AC^2 + BC^2$ $15^2 = 9^2 + BC^2$ $225 = 81 + BC^2$ $BC^2 = 144$ $BC = 12$

Now that we know $BC = 12$, we can calculate:

1. $\sin(A)$:

$\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AB} = \frac{12}{15} = \frac{4}{5}$

2. $\cot(A)$:

$\cot(A) = \frac{\text{adjacent}}{\text{opposite}} = \frac{AC}{BC} = \frac{9}{12} = \frac{3}{4}$

Thus: $\sin(A) = \frac{4}{5}, \quad \cot(A) = \frac{3}{4}$

Would you like more details or have any questions?

Here are five related questions:

1. How do we calculate $\cos(A)$ in a right triangle?
2. What is the value of $\tan(A)$?
3. Can you find $\sin(B)$ and $\cot(B)$ in this triangle?
4. How do you derive the Pythagorean identity from these values?
5. How would trigonometric values change if the triangle's sides were scaled?

Tip: For any right triangle, remember that the sine and cosine of an angle are simply the ratios of two sides, and these ratios do not depend on the actual size of the triangle, only on the angle itself.