Special Right Triangles

There are two special triangles. These triangles are special because their angles are used to often. 30-60-90, and 45-45-90.

From these two triangles we can come up with 6 important trig ratios that come up quite often.

Rendered by QuickLaTeX.com

 

How did we get those numbers?

A 45-45-90 is simply a square cut along its diagonal. If each side of the square was 1, then we’d use the Pythagorean Theorem to calculate the diagonal d.

    \begin{align*} 1^2+1^2&=d^2\\ 2&=d^2\\ d&=\sqrt{2} \end{align*}

Rendered by QuickLaTeX.com

A 30-60-90 triangle is an equilateral triangle, with side lengths of 2, cut by an altitude (segment from one angle, perpendicular to the opposite side). Again, two sides of the right triangle are known, so we apply the Pythagorean Theorem.\\

    \begin{align*} 1^2+&b^2=2^2\\ 1 + &b^2=4\\ &b^2=3\\ &b=\sqrt{3} \end{align*}

Rendered by QuickLaTeX.com