Right Triangles – Trigonometry Basics 1


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 180┬║. If one of these angles is 90┬║, 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.


About Todd Helmenstine

Todd Helmenstine is the physicist/mathematician who creates most of the images and PDF files found on sciencenotes.org. Nearly all of the graphics are created in Adobe Illustrator, Fireworks and Photoshop. Todd also writes many of the example problems and general news articles found on the site.


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