Sine and Cosine are very similar.............

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

Sine, Cosine and Tangent

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

They are often shortened to sin, cos and tan.

To calculate them:

Divide the length of one side by another side
... but which sides?

For a triangle with an angle θ, they are calculated this way:

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

In picture form:

Example: What is the sine of 35°?

Using this triangle (lengths are only to one decimal place):

sin(35°)	= Opposite / Hypotenuse
	= 2.8 / 4.9
	= 0.57...
cos(35°)	= Adjacent / Hypotenuse
	= 4.0 / 4.9
	= 0.82...
tan(35°)	= Opposite / Adjacent
	= 2.8 / 4.0
	= 0.70...

Practice Here:

How to remember? Think "Sohcahtoa"! It works like this:

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

You can read more about sohcahtoa ... please remember it, it may help in an exam !

Try It!

Have a try! Move the mouse around to see how different angles (in radians or degrees) affect sine, cosine and tangent.

In this animation the hypotenuse is 1, making the Unit Circle.

Notice that the adjacent side and opposite side can be positive or negative, which makes the sine, cosine and tangent change between positive and negative values also.

"Why didn't sin and tan go to the party?" "... just cos!"

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: 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!)