In a right-angled triangle ABC, right-angled at B, if AB = 3 cm and BC = 4 cm, find sinA and cosA.
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Step-by-Step Solution
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 sinA and cosA for angle A.
Step 2: Apply Pythagorean Theorem to Find Hypotenuse AC
To find sinA and cosA, 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 sinA, the opposite side BC is 4 cm and the hypotenuse AC is 5 cm. For cosA, the adjacent side AB is 3 cm and the hypotenuse AC is 5 cm.