Question

bc = \boxed{\space}
round your answer to the nearest hundredth.
right triangle abc with right angle at c, angle at b is 35 degrees, hypotenuse ab = 6, side bc is to be found

Explanation:

Step1: Identify the trigonometric ratio

In right triangle \(ABC\) (right - angled at \(C\)), we know the hypotenuse \(AB = 6\) and we want to find the length of \(BC\). The angle at \(B\) is \(35^{\circ}\). We use the cosine function, which is defined as \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\). For angle \(B\), the adjacent side to angle \(B\) is \(BC\) and the hypotenuse is \(AB\). So \(\cos(35^{\circ})=\frac{BC}{AB}\).

Step2: Solve for \(BC\)

We know that \(AB = 6\) and \(\theta=35^{\circ}\). Rearranging the formula \(\cos(35^{\circ})=\frac{BC}{6}\) to solve for \(BC\), we get \(BC = 6\times\cos(35^{\circ})\).
Using a calculator, \(\cos(35^{\circ})\approx0.8192\). Then \(BC=6\times0.8192 = 4.9152\).

Step3: Round to the nearest hundredth

Rounding \(4.9152\) to the nearest hundredth (two decimal places), we look at the third decimal place which is \(5\). When the third decimal place is \(5\) or more, we round up the second decimal place. So \(4.9152\approx4.92\).

Answer:

\(4.92\)