Tutor HuntResources Maths Resources
Memorising Trigonometric Functions And Identities
Date : 01/02/2016
Author Information
Uploaded by : Charanpal
Uploaded on : 01/02/2016
Subject : Maths
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.
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!
sin(A) = opposite/hypotenuse cos(A) = adjacent/hypotenuse tan(A) = opposite/adjacentHere, 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 Eulers formula:
eix = cos x + i sin xwhere e is the exponential function I is the unit imaginary number. We can then derive a number of useful equations:
eiy e-iy = 1 => cos2(y) + sin2(y) = 1
ei(y+z) = eiy eiz => 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!
This resource was uploaded by: Charanpal