Special Right Triangles 1


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.

The 30-60-90 triangle is a special right triangle.

This is an example of a special right triangle.

The 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.


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One thought on “Special Right Triangles

  • doug1943

    Every student should know the ratios for these two special right-angled triangles.

    An easy way to remember them is the mnemonic “Hee, hee, hee, one, two and the square root of three” for the 30-60 triangle, and “Boo, boo, boo, the square root of two” for the 45-45 triangle.