Activity 2.2.3.
In what follows, we work to understand key relationships in \(45^\circ\)-\(45^\circ\)-\(90^\circ\) and \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangles.
(a)
For the \(45^\circ\)-\(45^\circ\)-\(90^\circ\) triangle with legs of length \(x\) and \(y\) and hypotenuse of length \(1\text{,}\) what does the fact that the triangle is isosceles tell us about the relationship between \(x\) and \(y\text{?}\) What are their exact values?
(b)
Now consider the \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle with hypotenuse of length \(1\) and the longer leg (of length \(x\)) lying along the positive \(x\)-axis. What special kind of triangle is formed when we reflect this triangle across the \(x\)-axis? How can we use this perspective to determine the exact values of \(x\) and \(y\text{?}\)
(c)
Suppose we consider the related \(30^\circ\)-\(60^\circ\)-\(90^\circ\) triangle with hypotenuse of length \(1\) and the shorter leg (of length \(x\)) lying along the positive \(x\)-axis. What are the exact values of \(x\) and \(y\) in this triangle?
(d)
We know from the conversion factor from degrees to radians that an angle of \(30^\circ\) corresponds to an angle measuring \(\frac{\pi}{6}\) radians, an angle of \(45^\circ\) corresponds to \(\frac{\pi}{4}\) radians, and \(60^\circ\) corresponds to \(\frac{\pi}{3}\) radians.
Use your work in (a), (b), and (c) to label the noted point in each of the three respective figures with its exact coordinates.
