Question 1

Right-Angled Triangle: A triangle with one angle equal to 90°. The side opposite the right angle is the longest side, called the hypotenuse. The other two sides are called the legs (opposite and adjacent, depending on the angle in question).

Pythagoras Theorem: In a right-angled triangle, the square of the hypotenuse equals the sum of squares of the other two sides:

$AC^2 = AB^2 + BC^2$

Here, the right angle is at B, so AC is the hypotenuse. We first find AC, then use it to calculate the trigonometric ratios.

We will use the Pythagoras theorem and trigonometric ratios.

Step 1 — Find Hypotenuse AC

Let's find the length of the hypotenuse AC. We use the Pythagoras theorem.

$AC^2 = AB^2 + BC^2$

$AC^2 = (24\text{ cm})^2 + (7\text{ cm})^2$

$AC^2 = 576\text{ cm}^2 + 49\text{ cm}^2$

$AC^2 = 625\text{ cm}^2$

$AC = \sqrt{625\text{ cm}^2}$

$\boxed{AC = 25\text{ cm}}$

Step 2 — Determine sin A and cos A

For angle A, BC is the opposite side. AB is the adjacent side. AC is the hypotenuse.

$\sin A = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{BC}{AC}$

$\sin A = \frac{7}{25}$

$\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{AB}{AC}$

$\cos A = \frac{24}{25}$

Step 3 — Determine sin C and cos C

For angle C, AB is the opposite side. BC is the adjacent side. AC is the hypotenuse.

$\sin C = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AB}{AC}$

$\sin C = \frac{24}{25}$

$\cos C = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{BC}{AC}$

$\cos C = \frac{7}{25}$

Answer

(i) $\sin A = \frac{7}{25}$, $\cos A = \frac{24}{25}$ (ii) $\sin C = \frac{24}{25}$, $\cos C = \frac{7}{25}$