Question

In triangle $ABC$ if $\angle B = 90^\circ$, $AB = 5$ cm and $BC = 12$ cm, the value of $\sin A$ will be :

• A. 5/13
• B. 5/12
• C. 12/13
• D. 13/17

Hint:

5, 12, 13 is a very common Pythagorean triplet! Memorizing triplets like (3,4,5) and (5,12,13) saves a lot of calculation time.

Answer 1

The Correct Option is C

To find the value of \(\sin A\) in the given right triangle \(ABC\), we first need to apply the Pythagorean theorem and understand the trigonometric ratios. Given:

• \(\angle B = 90^\circ\)
• \(AB = 5 \text{ cm}\)
• \(BC = 12 \text{ cm}\)

First, apply the Pythagorean theorem to find the hypotenuse \(AC\):

\(AC^2 = AB^2 + BC^2\)

Substitute the given values:

\(AC^2 = 5^2 + 12^2 = 25 + 144 = 169\)

So, \(AC = \sqrt{169} = 13 \text{ cm}\).

Now, we know the sides of the triangle:

• \(AB = 5 \text{ cm}\)
• \(BC = 12 \text{ cm}\)
• \(AC = 13 \text{ cm}\)

In the right triangle \(ABC\), the sine of angle \(A\) is defined as the ratio of the length of the opposite side to the hypotenuse:

\(\sin A = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{BC}{AC} = \frac{12}{13}\)

Therefore, the correct answer is \(\frac{12}{13}\).