Basics of Trigonometry

Trigonometry (from Greek trigōnon, “triangle” and metron, “measure”) is a branch of mathematics that studies relationships between side lengths and angles of triangles.

A triangle has three sides and three angles.

The three angles always add to 180°.

There are three special names given to triangles that tell how many sides (or angles) are equal.



Note: In Formulae and Tables booklet lengths of sides of shapes are designated by lower case (small) letters, and angles are designated by upper case (capital) letters.



When a triangle has a right angle (90°) …

… and squares are made on each of the three sides, ..

… then the biggest square has the exact same area as the other two squares put together!


The longest side of the triangle is called the “hypotenuse”, so the formal definition is:

In a right angled triangle:
the square of the hypotenuse is equal to
the sum of the squares of the other two sides.

If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!)

Trigonometric functions relate an angle of a right-angled triangle to ratios of two side lengths.







If you have two triangles in which the angles in one are the same as the angles in the other, then the ratio of corresponding sides will be the same for each triangle.


Check out this site for nice simple and clear descriptions of different aspects of Trigonometry.

including the piece below on the Magic Hexagon.

Magic Hexagon for Trig Identities

This hexagon is a special diagram
to help you remember many trigonometric identities

Building It: The Quotient Identities

Start with:tan(x) = sin(x) / cos(x)(This one you just have to remember)
Then add:

  • cot (which is cotangent) opposite tan
  • csc (which is cosecant) next, and
  • sec (which is secant) last
To help you remember: the “co” functions are all on the right

OK, we have now built our hexagon, what do we get out of it?

Well, we can now follow “around the clock” (either direction) to get all the “Quotient Identities”:

Clockwise Anticlockwise
  • tan(x) = sin(x) / cos(x)
  • sin(x) = cos(x) / cot(x)
  • cos(x) = cot(x) / csc(x)
  • cot(x) = csc(x) / sec(x)
  • csc(x) = sec(x) / tan(x)
  • sec(x) = tan(x) / sin(x)
  • cos(x) = sin(x) / tan(x)
  • sin(x) = tan(x) / sec(x)
  • tan(x) = sec(x) / csc(x)
  • sec(x) = csc(x) / cot(x)
  • csc(x) = cot(x) / cos(x)
  • cot(x) = cos(x) / sin(x)

Product Identities

The hexagon also shows that a function between any two functions is equal to them multiplied together (if they are opposite each other, then the “1” is between them):

Example: tan(x)cos(x) = sin(x) Example: tan(x)cot(x) = 1

Some more examples:

  • sin(x)csc(x) = 1
  • tan(x)csc(x) = sec(x)
  • sin(x)sec(x) = tan(x)

But Wait, There is More!

You can also get the “Reciprocal Identities”, by going “through the 1”

Here you can see that sin(x) = 1 / csc(x)

Here is the full set:

  • sin(x) = 1 / csc(x)
  • cos(x) = 1 / sec(x)
  • cot(x) = 1 / tan(x)
  • csc(x) = 1 / sin(x)
  • sec(x) = 1 / cos(x)
  • tan(x) = 1 / cot(x)


AND we also get these:


  • sin(30°) = cos(60°)
  • tan(80°) = cot(10°)
  • sec(40°) = csc(50°)

Double Bonus: The Pythagorean Identities

The Unit Circle shows us that

sin2 x + cos2 x = 1

The magic hexagon can help us remember that, too, by going around any of these three triangles:

And we have:

  • sin2(x) + cos2(x) = 1
  • 1 + cot2(x) = csc2(x)
  • tan2(x) + 1 = sec2(x)

You can also travel backwards around a triangle. For example:

  • 1 – cos2(x) = sin2(x)

Trigonometry Tutorial from Engineers Ireland for Leaving Cert Maths. Lasts an hour and three quarters but you can always look at it in smaller bits at a time.


Trigonometry is one of the most important topics in Higher Level Leaving Cert maths. It is hard to avoid trigonometry, as well as accounting for questions in its own right, trigonometry is required in many other areas of the course, e.g., complex numbers, differentiation, integration and vectors.

Originally trigonometry was purely the study of triangles, but it has been expanded greatly since then. Our course reflects this development. We study triangles, sectors, circles, trigonometry ratios as well as the more general trigonometry functions. We learn how to prove trigonometry identities and how to solve trigonometry equations. We also examine inverse trigonometry functions and investigate a well-known trigonometry limit.