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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:

triangle showing Opposite, Adjacent and Hypotenuse
  • "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):

Opposite, Adjacent and Hypotenuse

Sine, Cosine and Tangent

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

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

calculator-sin-cos-tan

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):

30° triangle

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):

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

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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