Tag Archives: 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.

Right Triangles – Trigonometry Basics

The right triangle is a special type of triangle that forms the basis for nearly all trigonometry functions.

As you know, a triangle is a closed shape consisting of three sides. Every two of these sides form an angle between them for a total of three interior angles. These three angles together add up to 180º. If one of these angles is 90º, also known as a right angle, the triangle is known as a right triangle.

Right TriangleThis is a right triangle. The three angles A, B and C are marked and each side’s length is denoted by a, b, and c. The side length letters are the sides opposite of the angle with the corresponding capital letter. The side opposite the right angle C is known as the hypotenuse of the right triangle.

The length of the hypotenuse can be calculated using the Pythagorean Theorem

a2 + b2 = c2

Each triangle has six parts, 3 sides and 3 angles. If you know two of these values, you can calculate any of the other four using the six trigonometric ratios. These ratios are sine, cosine, tangent, cosecant, secant, and cotangent. Using the variables in the right triangle, the formulas for these ratios for angle A are:

Sine formula

cosine formula

tangent formula

cosecant formula

secant formula

tangent formula

The common method to remember these formulas is the mnemonic SOH CAH TOA.

S – Sine
O – Opposite
H – Hypotenuse

C – Cosine
A – Adjacent
H – Hypotenuse

T – Tangent
O – Opposite
A – Adjacent

The last three are reciprocals of the first three. The tricky part is to remember cosecant goes with sine, not cosine as the name may suggest.

Memorize and learn these relationships of a right triangle. You will see these formulas again as you further your studies in science and engineering.