# Right Angle Trigonometry

Back to “Basic Trigonometry” Page

“Trigonometry” means “measuring triangles.” Basic trigonometry refers to the relationships between the measures of angles and sides of right triangles on a flat plane.

You have already seen some of these relationships in our old friend the 3-4-5 right triangle.

Notice the Greek letter theta θ in the upper angle of the diagram. It doesn’t mean anything in particular. Math people use it because it looks cool. It refers to a trigonometry function.

## What is a Trig Function?

The acute angles of a right triangle can be expressed either in degrees or as a ratio of side lengths. For example, angle θ in the diagram above measures approximately 53°. But it can be expressed more precisely as a ratio of the opposite side to the hypotenuse or 4/5. This ratio is called the sine function. The three basic trig functions are sine, cosine, and tangent.

sine = opposite/hypotenuse

The sine of angle θ is the ratio of the opposite side over the hypotenuse, or 4 over 5.

The cosine of angle θ is the ratio of the adjacent side over the hypotenuse or 3/5.

The tangent of angle θ is the ratio of the opposite side over the adjacent, or 4/3.

## SOHCAHTOA

If you take the first letter of each of the bold-face words in the previous three sentences, you get the following word: sohcahtoa. This word can help you memorize the three basic trigonometry functions.

You will often find sine abbreviated to sin, cosine abbreviated to cos, and tangent abbreviated to tan.

## Reciprocals of Trig Functions

“Reciprocal” is a fancy word for flipping a fraction. Since the basic trig functions are ratios (fractions), they can be flipped upside down to make their reciprocals.

 The reciprocal of sine is cosecant (csc): csc = 1/sin The reciprocal of cosine is secant (sec): sec = 1/cos The reciprocal of tangent is cotangent (cot): cot = 1/tan

Consider the angle θ of the 3-4-5 triangle again.
If sin θ = 4/5, then csc θ = 5/4.
If cos θ = 3/5, then sec θ = 5/3.
If tan θ = 4/3, then cot θ = 3/4.