Question

ac =
round your answer to the nearest hundredth.
right triangle abc with right angle at c, angle at a is 40°, side ab = 4, find ac

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 adjacent side to the angle \(A = 40^{\circ}\), which is \(AC\). The cosine of an angle in a right triangle is defined as \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\). So, \(\cos(40^{\circ})=\frac{AC}{AB}\).

Step2: Solve for \(AC\)

We know that \(AB = 4\) and \(\theta=40^{\circ}\). From the formula \(\cos(40^{\circ})=\frac{AC}{4}\), we can solve for \(AC\) by multiplying both sides of the equation by \(4\). So, \(AC = 4\times\cos(40^{\circ})\).

We know that \(\cos(40^{\circ})\approx0.7660\) (using a calculator). Then \(AC=4\times0.7660 = 3.064\). Rounding to the nearest hundredth, we get \(AC\approx3.06\).

Answer:

\(3.06\)