One of the angles in a right triangle is \( \theta=46.0^{\circ} \), the hypotenuse has length 46.0 m . Find the length of the side adjacent to \( \theta: \square \) Find the length of the side opposite to \( \theta: \square \)

To find the lengths of the sides in a right triangle given \( \theta = 46.0^{\circ} \) and the hypotenuse \( h = 46.0 \) m, you can use the functions cosine and sine. The length of the side adjacent to \( \theta \) is calculated using the cosine function: \[ \text{Adjacent} = h \cdot \cos(\theta) = 46.0 \cdot \cos(46.0^{\circ}) \approx 46.0 \cdot 0.6947 \approx 31.99 \text{ m} \] The length of the side opposite to \( \theta \) is calculated using the sine function: \[ \text{Opposite} = h \cdot \sin(\theta) = 46.0 \cdot \sin(46.0^{\circ}) \approx 46.0 \cdot 0.7193 \approx 33.05 \text{ m} \] So, the lengths of the sides are approximately 31.99 m (adjacent) and 33.05 m (opposite). In ancient Greece, triangles and their properties captured the minds of mathematicians like Pythagoras and Euclid, leading to the formation of foundational concepts in geometry. The study of angles and sides not only assisted in architectural marvels but also laid the groundwork for trigonometry as a critical discipline, vital for navigation and astronomy. In practical applications, understanding the relationships between the angles and sides of triangles is crucial for fields such as engineering, physics, and architecture. For instance, when designing a ramp, architects must calculate the right angle and the length of each side to ensure safety and accessibility, demonstrating how these mathematical principles are integral to real-world structures.