Exact Values at 30, 45 and 60 Degrees Without a Formula Sheet

Only a few angles give clean, friendly exact values because only a few triangles have side ratios you can pin down without a calculator. Most angles produce messy decimals. But two familiar shapes, cut the right way, hand you exact ratios for free: an equilateral triangle sliced in half, and a square sliced along its diagonal. Everything in this topic comes from redrawing one of those two triangles.

Take an equilateral triangle with all sides equal to 2. Every angle is 60 degrees. Slice it straight down the middle from one vertex to the midpoint of the opposite side. That cut splits the 60-degree angle into two 30-degree angles and creates a right angle at the base. The original side of length 2 becomes the hypotenuse, the half-base becomes 1, and the height works out to √3 by the Pythagorean theorem. That gives the 30-60-90 triangle with sides 1, √3, 2 — short leg opposite 30 degrees, longer leg opposite 60 degrees, hypotenuse opposite the right angle. For the 45-45-90 triangle, start with a square of side 1 and cut along the diagonal. Each half is a right triangle with two legs of length 1 and a hypotenuse of √2, by the Pythagorean theorem again. Both angles besides the right angle are 45 degrees, so the triangle is isosceles — legs equal, hypotenuse √2.

These two triangles are the entire toolkit. Once an angle moves past 90 degrees, reference angles let you reuse the same ratios: the exact value keeps its size and only the sign changes based on the quadrant. That's the bridge to the unit circle, where these same right-triangle ratios become sine and cosine values attached to a point tracing a circle.