The special right triangle formula indicates that we get;
x = 4, y = 4·√3, z = 4·√6
The lengths of the sides of the triangle are found as follows;
The values of x, y, and z can be found using the side lengths, angles and trigonometric ratios of cosines sines and tangents as follows;
cos(60°) = x/8
x = 8 × cos(60°)
x = 4
Let h represent the height of the small right triangle we get;
sin(60°) = h/8
h = 8 × sin(60°)
8 × sin(60°) = 8 × √3/2
8 × √3/2 = 4·√3
h = 4·√3
The special 45° right triangle indicates that h = y
Therefore, y = 4·√3
sin(45°) = h/z
z = (4·√3)/sin(45°)
z = (4·√3)/(√2/2)
(4·√3)/(√2/2) = (4·√3) × (2/√2)
(4·√3) × (2/√2) = (8·√3)/√2
(8·√3)/√2 = (8·√3)/√2 × (√2/√2)
(8·√3)/√2 × (√2/√2) = 4·√6
z = 4·√6