Sine, Cosine and Tangent

Three Functions, but same idea.

Right Triangle

Sine, Cosine and Tangent are the main functions used in Trigonometry and are 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, Adjacent and Hypotenuse examples

Adjacent is always next to the angle

And Opposite is opposite the angle

Sine, Cosine and Tangent

Sine, Cosine and Tangent (often shortened to sin, cos and tan) are each a ratio of sides of a right angled triangle:

sin=opposite/hypotenuse cos=adjacent/hypotenuse tan=opposite/adjacent

For a given angle θ each ratio stays the same
no matter how big or small the triangle is

To calculate them:

Divide the length of one side by another side

Example: What is the sine of 35°?

triangle with 2.8, 4.0 and 4.9 sides

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...


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!

In picture form:

sin=opposite/hypotenuse cos=adjacent/hypotenuse tan=opposite/adjacent individual illustration

Practice Here:


How to remember? Think "Sohcahtoa"!

It works like this:

Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
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.

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



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:

30 degree triangle

Now we know the lengths, we can calculate the functions:

  sin(30°) = 1 / 2 = 0.5
  cos(30°) = 1.732 / 2 = 0.866...
  tan(30°) = 1 / 1.732 = 0.577...

(get your calculator out and check them!)

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

The classic 45° triangle has two sides of 1 and a hypotenuse of √2:

45 degree triangle

  sin(45°) = 1 / 1.414 = 0.707...
  cos(45°) = 1 / 1.414 = 0.707...
  tan(45°) = 1 / 1 = 1


Why are these functions important?

trig example

Example: Use the sine function to find "d"

We know:

And we want to know "d" (the distance down).

Start with:   sin 39° = opposite/hypotenuse
    sin 39° = d/30
Swap Sides:   d/30 = sin 39°
Use a calculator to find sin 39°:   d/30 = 0.6293…
Multiply both sides by 30:   d = 0.6293… x 30
    d = 18.88 to 2 decimal places.

The depth "d" is 18.88 m


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.

You can also see Graphs of Sine, Cosine and Tangent.

Less Common Functions

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

They are equal to 1 divided by cos, 1 divided by sin, and 1 divided by tan:

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