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:
(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):
|
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):
|
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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