Sine, Cosine and Tangent
Three Functions, but same idea.
Right Triangle
Sine, Cosine and Tangent are all based on a Right-Angled Triangle
Before getting stuck into the functions, it helps to give a name to each side of a right triangle:
- "Opposite" is opposite to the angle θ
- "Adjacent" is adjacent (next to) to the angle θ
- "Hypotenuse" is the long one
Adjacent is always next to the angle (and opposite is opposite the angle):
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Sine, Cosine and Tangent
The three main functions in trigonometry are Sine, Cosine and Tangent.
| They are often shortened to sin, cos and tan. |
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"Why didn't sin and tan
go to the party?"
"... just cos!" |
|
The calculation is simply one side divided by another side ... you just have to know which sides!
For a triangle with an angle θ , the functions are calculated this way
Sine Function: |
sin(θ) = Opposite / Hypotenuse |
Cosine Function: |
cos(θ) = Adjacent / Hypotenuse |
Tangent Function: |
tan(θ) = Opposite / Adjacent |
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Good calculators have sin, cos and tan on them, to make it easy for you. Just put in the angle and press the button.
But you still need to remember what they mean! |
Examples
Example 1: what are the sine, cosine and tangent of 30° ?
The classic 30° triangle has a hypotenuse of length 2, an opposite side of length 1 and an adjacent side of
√(3):
Now we know the lengths, we can calculate the functions:
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 them!)
Example 2: what are 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 |
You can read more about sohcahtoa ...
... but please remember "sohcahtoa" - it could help in an exam !
Why?
Why are these functions important?
- Because they let you work out angles when you know sides
- And they let you work out sides when you know angles
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) |
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