In a right triangle the two legs are 3 cm and 4 cm. Find the hypotenuse and sin of the angle opposite the 3 cm side.

Step 1: Identify given values and apply Pythagorean theorem

We are given the lengths of the two legs of a right triangle, $a = 3 \text{ cm}$ and $b = 4 \text{ cm}$. To find the hypotenuse, $c$, we use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Step 2: Calculate the hypotenuse

Substitute the given leg lengths into the Pythagorean theorem. Calculate the squares of the legs, sum them, and then take the square root to find the length of the hypotenuse.

Step 3: Identify the angle and apply sine definition

We need to find the sine of the angle opposite the 3 cm side. Let this angle be $\theta$. The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

Step 4: Calculate the sine of the angle

For the angle opposite the 3 cm side, the opposite side is 3 cm and the hypotenuse is 5 cm. Substitute these values into the sine formula.

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