Question

ac = \boxed{ } round your answer to the nearest hundredth. triangle with right angle at c, angle at a is 40°, side ab = 4, ac is unknown (marked ?)

Explanation:

Step1: Identify the trigonometric ratio

We have a right triangle \( \triangle ABC \) with \( \angle C = 90^\circ \), \( \angle A = 40^\circ \), and hypotenuse \( AB = 4 \). We need to find the length of \( AC \), which is adjacent to \( \angle A \). The cosine function relates the adjacent side and the hypotenuse: \( \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} \). So, \( \cos(40^\circ)=\frac{AC}{AB} \).

Step2: Substitute the known values

We know \( AB = 4 \) and \( \theta = 40^\circ \). Substituting these into the cosine formula: \( \cos(40^\circ)=\frac{AC}{4} \). To solve for \( AC \), we multiply both sides by 4: \( AC = 4 \times \cos(40^\circ) \).

Step3: Calculate the value

Using a calculator to find \( \cos(40^\circ) \approx 0.7660 \). Then, \( AC = 4 \times 0.7660 = 3.064 \). Rounding to the nearest hundredth, we get \( AC \approx 3.06 \).

Answer:

\( 3.06 \)