In a right-angled triangle △ ABC, the right angle is at B. If AB = 24 cm and BC = 7 cm, find the values of sinA, cosA, sinC, and cosC.
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Step-by-Step Solution
Step 1: Find the hypotenuse using the Pythagorean theorem
In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This is known as the Pythagorean theorem. Here, AC is the hypotenuse.
Step 2: Calculate the length of the hypotenuse
Substitute the given lengths of AB=24 cm and BC=7 cm into the Pythagorean theorem. Calculate the squares, sum them, and then take the square root to find the length of AC.
Step 3: Calculate sinA and cosA
For angle A, the side opposite is BC and the side adjacent is AB. The hypotenuse is AC. We use the definitions of sine (Opposite/Hypotenuse) and cosine (Adjacent/Hypotenuse) to find their values.
Step 4: Calculate sinC and cosC
For angle C, the side opposite is AB and the side adjacent is BC. The hypotenuse remains AC. We apply the same definitions of sine and cosine, but with respect to angle C.