To find the length of segment AC in triangle ABC, we can use the properties of right triangles and trigonometric functions. We have the following information:
- Triangle ABC is a right triangle with angle B as the right angle.
- Angle CAB = 34 degrees.
- Segment AB = 3 units.
First, we will find the length of segment AC using the sine function.
Since angle CAB is opposite to segment BC, we can use the sine:
\[ \sin(\text{angle CAB}) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AC} \]
We do not yet know BC, but we can express BC in terms of AC. Using the cosine function for angle CAB, we have:
\[ \cos(\text{angle CAB}) = \frac{AB}{AC} \]
We know that AB = 3, so we can write:
\[ \cos(34^\circ) = \frac{3}{AC} \]
Now solving for AC yields:
\[ AC = \frac{3}{\cos(34^\circ)} \]
Calculating \(\cos(34^\circ)\) using a calculator:
\[ \cos(34^\circ) \approx 0.829 \]
Substituting this value in:
\[ AC \approx \frac{3}{0.829} \approx 3.62 \]
So the length of segment AC, rounded to the nearest hundredth, is approximately 3.62 units.
Therefore, the correct choice is:
3.62 units.
1 answer