Question

in right - triangle abc, angle c = 90 degrees, angle a = 65 degrees, and the hypotenuse ab = 16 cm. find the length of side bc (denoted as x). options: 6.8 cm, 7.5 cm, 14.5 cm, 17.7 cm

Explanation:

Step1: Identify the trigonometric relation

In right - triangle \(ABC\) with right - angle at \(C\), we know the hypotenuse \(AB = 16\mathrm{cm}\) and we want to find the side \(x\) (opposite to angle \(A = 65^{\circ}\)). We use the sine function. The sine of an angle in a right - triangle is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), so \(\sin A=\frac{BC}{AB}\).

Step2: Substitute the values

We know that \(A = 65^{\circ}\) and \(AB = 16\mathrm{cm}\). So, \(\sin65^{\circ}=\frac{x}{16}\). Since \(\sin65^{\circ}\approx0.9063\), we can solve for \(x\) by multiplying both sides of the equation by \(16\): \(x = 16\times\sin65^{\circ}\).

Step3: Calculate the value of \(x\)

\(x=16\times0.9063 = 14.5008\approx14.5\mathrm{cm}\)

Answer:

14.5 cm