Special Right Triangles
Key Questions

Answer:
Consider the properties of the sides, the angles and the symmetry.
Explanation:
#454590" "# refers to the angles of the triangle.The
#color(blue)("sum of the angles is " 180°)# There are
#color(blue)("two equal angles")# , so this is an isosceles triangle.It therefore also has
#color(blue)(" two equal sides.")# The third angle is
#90°# . It is a#color(blue)("rightangled triangle")# therefore Pythagoras' Theorem can be used.The
#color(blue)("sides are in the ratio " 1 :1: sqrt2)# It has
#color(blue)("one line of symmetry")#  the perpendicular bisector of the base (the hypotenuse) passes through the vertex, (the#90°# angle).It has
#color(blue)("no rotational symmetry.")# 
#mathbf{30^circ""60^circ""90^circ}# TriangleThe ratios of three sides of a
#30^circ""60^circ""90^circ# triangle are:#1:sqrt{3}:2#
I hope that this was helpful.

Each blackandred (or blackandyellow) triangles is a special rightangled triangle. The figures outside the circle 
#pi/6, pi/4, pi/3#  are the angles that the triangles make with the horizontal (x) axis. The other figures #1/2, sqrt(2)/2, sqrt(3)/2#  are the distances along the axes  and the answers to#sin(x)# (yellow) and#cos(x)# (red) for each angle. 

#30^circ# #60^circ# #90^circ# Triangles whose sides have the ratio#1:sqrt{3}:2# 
#45^circ# #45^circ# #90^circ# Triangles whose sides have the ratio#1:1:sqrt{2}#
These are useful since they allow us to find the values of trigonometric functions of multiples of
#30^circ# and#45^circ# . 