Question

solve for x. round to the nearest tenth, if necessary.

(image of a right triangle abc with right angle at b, side ab = 8, angle at c is 70 degrees, and side bc = x)

answer attempt 1 out of 2
x = input box 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 \(C = 70^{\circ}\). We can use the cosine function. The cosine of an angle in a right triangle is defined as \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\). For angle \(C = 70^{\circ}\), the adjacent side is \(x\) and the hypotenuse is \(8\). 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 \(\cos(70^{\circ})\approx0.3420\) (using a calculator). Then \(x=8\times0.3420 = 2.736\). Rounding to the nearest tenth, we get \(x\approx2.7\).

Answer:

\(x\approx2.7\)