Memorising Trigonometric Functions And Identities

Part of the GCSE and A-level mathematics syllabus covers trigonometric identities and formulae which can be difficult to remember so here are some mnemonics to help you. Firstly, there are the three definitions of the main trigonometric functions for a right angled triangle.

sin(A) = opposite/hypotenuse

cos(A) = adjacent/hypotenuse

tan(A) = opposite/adjacent

Here, A is the angle, opposite is the length of opposite side to the angle, adjacent is the side adjacent to the angle but forming the right angle, and the hypotenuse links the two other sides. These can be remembered using the mnemonic: Some Old Houses Can Always Hide Their Old Age.

From these equations we can derive the relationship between the three:

tan(A) = opp / adj

= (sin(A) * hyp) / (cos(A) * hyp)

= sin(A) / cos(A)

which is an important equation to remember. We have abbreviated opposite, adjacent and hypotenuse as opp, add and hyp respectively.

Next, we come to some trigonometric identities. A good place to start with this is simply to memorise Euler’s formula:

eⁱx = cos x + i sin x

where e is the exponential function I is the unit imaginary number. We can then derive a number of useful equations:

e^iy e^-iy = 1

=> cos²(y) + sin²(y) = 1

e^i(y+z) = e^iy e^iz

=> cos(y+z) + i sin(y+z) = (cos(y) + i sin(y))(cos(z) + i sin(z))

= cos(y) cos(z) + cos(y) i sin(z) + i sin(y) cos(z) – sin(y) sin(z)

=> cos(y + z) = cos(y) cos(z) – sin(y) sin(z)

sin(y + z) = cos(y) sin(z) + sin(y) cos(z)

tan(y + z) = (cos(y) sin(z) + sin(y) cos(z))/(cos(y) cos(z) – sin(y) sin(z))

= (tan(z) + tan(y))/(1 – tan(y) tan (z))

For these it is easy to derive the double angle identities:

sin(2y) = cos(y) sin(y) + sin(y) cos(y) = 2 sin(y) cos(y)

cos(2y) = cos (y) cos(y) – sin(y) sin(y) = cos² (y) – sin² (y) = 2 cos²(y) – 1

tan(2y) = 2 tan(y)/(1 - tan²(y))

Hope that helps!