Tag Archives: special right triangles

Special Right Triangles

Right triangles are central to trigonometry. Two special right triangles appear over and over in standardized exams and homework problems. These triangles are “special” because they have simple ratios between the lengths of each side.

30-60-90 Right TriangleThe first is the 30-60-90 triangle.

The 30-60-90 is named after the interior angles of the triangle. If the shortest side is length x, the longest side or hypotenuse is twice as long. The remaining leg of the triangle is √3 times the length of the short leg.

The trigonometric ratios for these angles are easy to figure out. Let’s look at the 30º angle:

sin30
cos30
tan30
csc30
sec30
cot30

45-45-90 Right TriangleThe second special triangle is the 45-45-90 triangle. This triangle is formed by cutting a square along its diagonal. If the sides of the square are length x, the hypotenuse will be x√2.

Both angles are 45º, so the trigonometric ratios are the same for both interior angles.

sin45
cos45
tan45
csc45
sec45
cot45

Memorizing these values would be time well spent. These triangles will show up again and again in exams and homework.

For more general right triangles, check out Right Triangle Trigonometry.