Question

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Explanation:

Let's solve problem 12: a right - triangle with a 60° angle, hypotenuse = 10, find \(x\) (opposite to 60°) and \(y\) (adjacent to 60°).

Step 1: Recall trigonometric ratios for 30 - 60 - 90 triangle

In a 30 - 60 - 90 right - triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite 30° is the shortest one (let's call it \(s\)), the side opposite 60° is \(s\sqrt{3}\), and the hypotenuse is \(2s\).

First, we need to determine which angle is 30°. Since one angle is 90° and another is 60°, the third angle is \(180^{\circ}-90^{\circ}-60^{\circ}=30^{\circ}\). The side adjacent to 60° ( \(y\)) is opposite to 30°, and the side opposite to 60° ( \(x\)) is adjacent to 30°. The hypotenuse \(h = 10\).

Step 2: Find the length of \(y\) (adjacent to 60°, opposite to 30°)

For a 30 - 60 - 90 triangle, the side opposite 30° (\(y\)) is half of the hypotenuse. So, if the hypotenuse \(h = 10\), then \(y=\frac{h}{2}\)
\(y=\frac{10}{2}=5\)

Step 3: Find the length of \(x\) (opposite to 60°, adjacent to 30°)

The side opposite 60° (\(x\)) is \(\frac{\sqrt{3}}{2}\) times the hypotenuse. So \(x = h\times\frac{\sqrt{3}}{2}\)
Substitute \(h = 10\) into the formula: \(x=10\times\frac{\sqrt{3}}{2}=5\sqrt{3}\)

Answer:

\(x = 5\sqrt{3}\), \(y = 5\)