Sure, let's solve this problem step-by-step.
1. Understand the Problem: You are given a right triangle with hypotenuse 13 and one of the angles is [tex]\(3^\circ\)[/tex]. You need to find the length of the adjacent side to this angle.
2. Recall Definitions:
- The hypotenuse is the longest side of a right triangle, opposite the right angle.
- The adjacent side is the side next to the angle in question.
- The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse: [tex]\( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)[/tex].
3. Set Up the Equation: Let [tex]\( \theta \)[/tex] be [tex]\(3^\circ\)[/tex] and the hypotenuse be 13. We can write the formula for cosine:
[tex]\[
\cos(3^\circ) = \frac{\text{adjacent}}{13}
\][/tex]
4. Solve for the Adjacent Side: Rearrange the formula to solve for the adjacent side:
[tex]\[
\text{adjacent} = 13 \cdot \cos(3^\circ)
\][/tex]
5. Calculate the Value: By computing this expression, you find:
[tex]\[
\text{adjacent} \approx 12.98218395180946
\][/tex]
6. Summary: Therefore, the length of the adjacent side, rounded to the nearest tenth, is approximately [tex]\(13.0\)[/tex].
The problem involves finding the length of the adjacent side in a right triangle, given an angle and the hypotenuse, and uses the cosine ratio to do so. The result is approximately 13.0 when rounded to the nearest tenth.
