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$=/sqrt(2)$ $=/sqrt(3)$ √2 √3

Section 20.2 - 30-60-90 Triangles

30-60-90 triangles have a different length ratio--1:square root of three:2. This is confirmed by the Pythagorean theorem as well: 1^2+3=2^2. Clearly, the smallest side is opposite the smallest angle, so for example, in triangle ABC, with angles A, B, and C having measure 30, 60, and 90 degrees respectively and AB having length 1, BC will have length 1/2 and AC will have length sqrt(3)/2, or 0.866...

Of course, these triangles could be solved by trigonometry, but these ratios provide a shortcut. In fact, they help us remember the most important trigonometric values in the 0-to-90 degree range:

sin(0)=0   sin(30)=1/2        sin(45)=sqrt(2)/2, or 1/sqrt(2)  sin(60)=sqrt(3)/2   sin(90)=1

cos(0)=1   cos(30)=sqrt(3)/2  cos(45)=sqrt(2)/2, or 1/sqrt(2)  cos(60)=1/2         cos(90)=0

tan(0)=0   tan(30)=sqrt(3)/3  tan(45)=1                        tan(60)=sqrt(3)     tan(90) is not defined.

Note that sine divided by cosine equals tangent, and also that

sin(90-x)=cos x, cos(90-x)=sin x, and tan(90-x)=1/tan x.