# User:Chief Mike/Sandbox

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√2 √3

## Section 20.2 - 30-60-90 Triangles[edit]

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.