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Sine, Cosine and Tangent

Three Functions, but same idea.

Right Triangle

Before getting stuck into the functions, it helps to give the three sides of a right triangle names, as follows:

triangle showing Opposite, Adjacent and Hypotenuse
(Adjacent is adjacent to the angle, and Opposite is opposite ... of course!)

Sine, Cosine and Tangent

The three main functions in trigonometry are Sine, Cosine and Tangent.

They are often shortened to just "sin", "cos" and "tan".

How to Calculate

They can be calculated by dividing the length of one side by another ... you just have to know which sides!

In relation to the angle θ :

Sine Function:
sin(θ) = Opposite / Hypotenuse
Cosine Function:
cos(θ) = Adjacent / Hypotenuse
Tangent Function:
tan(θ) = Opposite / Adjacent

Examples

Example 1: what is the sine, cosine and tangent of 30° ?

The classic 30° triangle has a hypotenuse of length 2, an opposite side of length 1 and a adjacent side of √(3):
30° triangle

Sine
sin(30°) = 1 / 2 = 0.5
Cosine
cos(30°) = 1.732 / 2 = 0.866
Tangent
tan(30°) = 1 / 1.732 = 0.577
(get your calculator out and check it!)

Example 2: what is the sine, cosine and tangent of 45° ?

The classic 45° triangle has two sides of 1 and a hypotenuse of √(2):
45° triangle
Sine
sin(45°) = 1 / 1.414 = 0.707
Cosine
cos(45°) = 1 / 1.414 = 0.707
Tangent
tan(45°) = 1 / 1 = 1

 

Sohcahtoa

Sohca...what? Just an easy way to remember which side to divide by which! Like this:

Soh...
Sine = Opposite / Hypotenuse
...cah...
Cosine = Adjacent / Hypotenuse
...toa
Tangent = Opposite / Adjacent

Remember "sohcahtoa" - it could help in an exam !

Exercise

Try this paper-based exercise where you can calculate the sine function for all angles from 0° to 360°, and then graph the result. It will help you to understand these relatively simple functions.

Less Common Functions

To complete the picture, there are 3 other functions where you divide one side by another, but they are not so commonly used.

They are equal to the 1 divided by each of the three main functions (sin, cos and tan), like this:

Secant Function:
sec(θ) = Hypotenuse / Adjacent (=1/cos)
Cosecant Function:
csc(θ) = Hypotenuse / Opposite (=1/sin)
Cotangent Function:
cot(θ) = Adjacent / Opposite (=1/tan)

And that is all!

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