91Ó°ÊÓ

A right triangle is a triangle in which one angle is exactly \(90^{\text{°}}\). This defines a specific relationship between its sides through trigonometric functions and the Pythagorean theorem. In a right triangle, the side opposite the \(90^{\text{°}}\) angle is the hypotenuse, the longest side of the triangle. The other two sides are called the legs. Thus, in the scenario where the ship's path forms a right triangle with the lighthouse, the legs of the right triangle are the \(1800\) meters the ship traveled along its path and the distance from the ship to the lighthouse at the second sighting. The hypotenuse is the initial distance from the ship to the lighthouse when first seen at a \(42^{\text{°}}\) angle.

Trigonometric functions relate angles of a triangle to the lengths of its sides. The primary functions include sine, cosine, and tangent. In our problem, the sine function is used. The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. Therefore, \text{sin}(angle) = \(\frac{opposite}{hypotenuse}\). In the context of this exercise, to find the distance from the ship to the lighthouse at the beginning, we used the sine function: \text{sin}(42^{\text{°}}) = \(\frac{\text{distance BC}}{\text{distance AC}}\). We'd then rearrange this to find the hypotenuse (distance AC).

The Pythagorean theorem is a fundamental principle of geometry that applies to right triangles. It states that the square of the hypotenuse (the side opposite the \(90^{\text{°}}\) angle) is equal to the sum of the squares of the other two sides: \text{hypotenuse}\(^{2}\) = \text{leg}\(^{2}\) + \text{leg}\(^{2}\). In our problem involving the ship and the lighthouse, after finding the hypotenuse (AC), we use the Pythagorean theorem to solve for the distance from the ship to the lighthouse at the second sighting (BC): AC\(^{2}\) = AB\(^{2}\) + BC\(^{2}\). By isolating BC and solving, we calculate this distance directly.

The sine function is one of the basic trigonometric functions that relates an angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse. Formally, \text{sin}(theta) = \(\frac{opposite}{hypotenuse}\). This principle is used to solve trigonometric problems like in the given exercise. To determine the distance AC, where the angle is \(42^{\text{°}}\), the sine function is essential: \text{sin}(42^{\text{°}}) = \(\frac{BC}{AC}\). By rearranging, we get: AC = \(\frac{AB}{\text{sin}(42^{\text{°}})}\). Here, AB is \(1800\) meters, and \text{sin}(42^{\text{°}})\(\text{ is approximately }0.6691\). Therefore, AC = \(\frac{1800}{0.6691}\) ≈ \(2690.6\) meters.