In a right-angled triangle ABC, right-angled at B, if AB = 3 cm and BC = 4 cm, find $\sin A$ and $\cos A$.

Step 1: Identify Given Information and Goal

We are given a right-angled triangle ABC, with the right angle at B. The lengths of two sides, AB and BC, are provided. Our goal is to determine the values of $\sin A$ and $\cos A$ for angle A.

Step 2: Apply Pythagorean Theorem to Find Hypotenuse AC

To find $\sin A$ and $\cos A$, we need the lengths of all three sides of the triangle, especially the hypotenuse. We can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Here, AC is the hypotenuse.

Step 3: Substitute Values and Calculate Hypotenuse

Now we substitute the given lengths of AB = 3 cm and BC = 4 cm into the Pythagorean theorem equation. After calculating the squares and summing them, we take the square root to find the length of the hypotenuse AC. The length of the hypotenuse AC is 5 cm.

Step 4: Define Sine and Cosine for Angle A

In a right-angled triangle, the sine of an angle is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse. The cosine of an angle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. For angle A, BC is the opposite side and AB is the adjacent side.

Step 5: Calculate Sine A and Cosine A

Using the definitions, we substitute the calculated side lengths into the formulas. For $\sin A$, the opposite side BC is 4 cm and the hypotenuse AC is 5 cm. For $\cos A$, the adjacent side AB is 3 cm and the hypotenuse AC is 5 cm.

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• (secθ - tanθ) = (1)/(m) --- (2) Adding (1) & (2) secθ + tanθ + secθ - tanθ = (1)/(m) + (m)/(1) 2secθ = (1 + m²)/(m) secθ = (1 + m²)/(2m) (Hence proved) 2) If as
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