Question

c = \boxed{} und your answer to the nearest hundredth. diagram: right triangle ( abc ) with right angle at ( c ), ( angle b = 70^circ ), side ( ab = 4 ), side ( ac ) (opposite ( angle b )) marked with ( ? ).

Explanation:

Step1: Identify the trigonometric ratio

In right triangle \(ABC\) (right - angled at \(C\)), we know the hypotenuse \(AB = 4\) and we want to find the length of \(AC\). The angle at \(B\) is \(70^{\circ}\). We use the cosine function because \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For angle \(B = 70^{\circ}\), the adjacent side to angle \(B\) is \(BC\)? No, wait, the adjacent side to angle \(B\) for finding \(AC\)? Wait, no. Wait, angle at \(B\) is \(70^{\circ}\), the side \(AC\) is opposite to angle \(B\)? Wait, no. Wait, right - angled at \(C\), so \(\angle C = 90^{\circ}\), \(\angle B=70^{\circ}\), hypotenuse \(AB = 4\). We want to find \(AC\). The sine of angle \(B\) is \(\sin B=\frac{AC}{AB}\), because \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, the side opposite to angle \(B\) is \(AC\), and the hypotenuse is \(AB = 4\).

Step2: Apply the sine formula

We know that \(\sin B=\frac{AC}{AB}\), so \(AC = AB\times\sin B\). Given \(AB = 4\) and \(B = 70^{\circ}\), we have \(AC=4\times\sin(70^{\circ})\).

We know that \(\sin(70^{\circ})\approx0.9397\). Then \(AC = 4\times0.9397 = 3.7588\approx3.76\) (rounded to the nearest hundredth).

Answer:

\(3.76\)