In a right-angled triangle △ ABC, the right angle is at B. If AB = 24 cm and BC = 7 cm, find the values of $\sin A$, $\cos A$, $\sin C$, and $\cos C$.

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 $AB = 24 \text{ cm}$ and $BC = 7 \text{ cm}$ into the Pythagorean theorem. Calculate the squares, sum them, and then take the square root to find the length of $AC$.

Step 3: Calculate $\sin A$ and $\cos A$

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 $\sin C$ and $\cos C$

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 $C$.

More from this chapter

• **EXERCISE 8.3** **1.** Express the trigonometric ratios sin A, tan A and sec A in terms of cot A.
• Cos A / 1 + sin a + 1 + sin a / cos A is equal to 2 sec a
• write all the other trigonometric ratios of angle A in terms of sec A
• Express all the trigonometric ratios in terms of Sin A
• Express the trigonometric ratio sign a sec a tan a in terms of cot a
• Prove that √(sec² A + csc² A) = tan A + cot A
• (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
• If tanθ + secθ = m, prove that secθ = (m² + 1)/(2m).

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