Question

solve for x. round to the nearest tenth, if necessary.
(image of triangle abc with right angle at b, side ab = 8, angle at c is 70 degrees, side bc = x)
answer attempt 2 out of 2
x = |
submit answer

Explanation:

Step1: Identify the trigonometric ratio

In right triangle \(ABC\) (right - angled at \(B\)), we know the hypotenuse \(AC = 8\) and we want to find the length of the side \(BC=x\) which is opposite to the angle \(A\) and adjacent to the angle \(C = 70^{\circ}\). Wait, actually, angle \(C = 70^{\circ}\), side \(AB\) is adjacent to angle \(C\)? No, let's re - identify. In right - triangle \(ABC\) with \(\angle B = 90^{\circ}\), \(\angle C=70^{\circ}\), hypotenuse \(AC = 8\), and we want to find \(BC=x\). The sine of an angle in a right - triangle is defined as \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\) and cosine is \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\). For angle \(C = 70^{\circ}\), the side opposite to angle \(C\) is \(AB\) and the side adjacent to angle \(C\) is \(BC=x\), and the hypotenuse is \(AC = 8\). Wait, no, \(\sin(C)=\frac{AB}{AC}\) and \(\cos(C)=\frac{BC}{AC}\). So \(\cos(70^{\circ})=\frac{x}{8}\).

Step2: Solve for \(x\)

We can re - arrange the formula \(\cos(70^{\circ})=\frac{x}{8}\) to solve for \(x\). Multiply both sides of the equation by \(8\): \(x = 8\times\cos(70^{\circ})\).
We know that \(\cos(70^{\circ})\approx0.3420\). Then \(x=8\times0.3420 = 2.736\approx2.7\) (rounded to the nearest tenth).

Answer:

\(2.7\)