Question

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Explanation:

Step1: Identify the trigonometric ratio

In right triangle \(ABC\) with right angle at \(C\), we know angle \(A = 40^\circ\), side \(BC\) is opposite to angle \(A\)? Wait, no, wait. Wait, \(AC\) is adjacent to angle \(A\), and \(AB\) is the hypotenuse? Wait, no, the side labeled 7 is \(AC\)? Wait, no, the diagram: right angle at \(C\), so \(AC\) and \(BC\) are legs, \(AB\) is hypotenuse. Wait, angle at \(A\) is \(40^\circ\), and the side adjacent to angle \(A\) is \(AC\), and the hypotenuse is \(AB = 4\)? Wait, no, the length labeled 4 is \(AB\), and the side \(AC\) is one leg, length 7? Wait, no, maybe I misread. Wait, the triangle: right angle at \(C\), so \(\angle C = 90^\circ\), \(\angle A = 40^\circ\), \(AB = 4\)? Wait, no, the number 4 is on \(AB\), and 7 is on \(AC\)? Wait, no, that can't be, because hypotenuse should be longer than legs. Wait, maybe the side with length 7 is \(AC\), and we need to find \(AC\)? Wait, no, the problem says \(AC =\) [blank], and we have angle \(A = 40^\circ\), hypotenuse \(AB = 4\)? Wait, no, that would make \(AC = AB \times \cos(40^\circ)\). Wait, let's clarify:

In right triangle \(ABC\), \(\angle C = 90^\circ\), \(\angle A = 40^\circ\), hypotenuse \(AB = 4\)? Wait, no, the length 7 is on \(AC\)? Wait, maybe the diagram has \(AC = 7\)? No, the problem says "Round your answer to the nearest hundredth", so we need to calculate \(AC\) using trigonometry. Let's assume that \(AB\) is the hypotenuse with length 4, and \(\angle A = 40^\circ\), and we need to find \(AC\) (adjacent side to angle \(A\)). Then, \(\cos(\angle A) = \frac{AC}{AB}\), so \(AC = AB \times \cos(40^\circ)\). Wait, but if \(AB = 4\), then \(AC = 4 \times \cos(40^\circ)\). Wait, but \(\cos(40^\circ) \approx 0.7660\), so \(4 \times 0.7660 \approx 3.06\). But that contradicts if \(AC\) is a leg and hypotenuse is 4, but 3.06 is less than 4, which is okay. Wait, maybe the length 4 is \(AB\), and we need to find \(AC\) using \(\cos(40^\circ)\).

Wait, let's re-express:

\(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\)

Here, \(\theta = 40^\circ\), adjacent side is \(AC\), hypotenuse is \(AB = 4\). So:

\(\cos(40^\circ) = \frac{AC}{4}\)

So, \(AC = 4 \times \cos(40^\circ)\)

Step2: Calculate the value

First, find \(\cos(40^\circ)\). Using a calculator, \(\cos(40^\circ) \approx 0.7660444431\)

Then, \(AC = 4 \times 0.7660444431 \approx 3.064177772\)

Round to the nearest hundredth: 3.06

Wait, but wait, maybe the side with length 7 is \(BC\) (opposite to angle \(A\)), and we need to find \(AC\) (adjacent). Then, \(\tan(40^\circ) = \frac{BC}{AC}\), so \(AC = \frac{BC}{\tan(40^\circ)}\). If \(BC = 7\), then \(AC = \frac{7}{\tan(40^\circ)}\). \(\tan(40^\circ) \approx 0.8391\), so \(7 / 0.8391 \approx 8.34\). But that would make hypotenuse \(AB = \sqrt{7^2 + 8.34^2} \approx \sqrt{49 + 69.55} \approx \sqrt{118.55} \approx 10.89\), which is longer than 4. So maybe my initial assumption is wrong.

Wait, the diagram: right angle at \(C\), so \(AC\) and \(BC\) are legs, \(AB\) is hypotenuse. The label 4 is on \(AB\), and 7 is on \(AC\)? No, that can't be, because hypotenuse must be longer than legs. So maybe the number 4 is on \(BC\) (opposite to angle \(A\)), and we need to find \(AC\) (adjacent). Then, \(\tan(40^\circ) = \frac{BC}{AC}\), so \(AC = \frac{BC}{\tan(40^\circ)}\). If \(BC = 4\), then \(AC = \frac{4}{\tan(40^\circ)} \approx \frac{4}{0.8391} \approx 4.77\). But the problem says "Round your answer to the nearest hundredth".

Wait, maybe I misread the diagram. Let's check again: the triangle has right angle at \(C\), \(\angle A = 40^\circ\), \(AB = 4\) (hypotenuse), and we need to find \(AC\) (adjacent to \(\angle A\)). Then:

\(\cos(40^\circ) = \frac{AC}{AB}\)

\(AC = AB \times \cos(40^\circ)\)

\(AB = 4\), so \(AC = 4 \times \cos(40^\circ) \approx 4 \times 0.7660 \approx 3.06\)

Yes, that makes sense. So the steps are:

1. Identify the trigonometric ratio: cosine, since we have hypotenuse and need adjacent side.

2. Apply the formula: \(AC = AB \times \cos(40^\circ)\)

3. Calculate the value using a calculator.

Answer:

\(3.06\)