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Trigonometric Functions

Before you begin factorising make sure you understand some differences in notation.

2sin x means "obtain the sine of x then double the result" whereas
sin 2x means "double x and then obtain the sine of twice x"
sin2 x means "obtain the sine of x then square the result" whereas
sin x2 means "square x then obtain the sine of x squared".

Generally these four expressions are not equivalent!

Example 4


xcos2 x + 4x2 cos x sin x


xcos2 x and 4x2 cos x sin x

Consider each product separately, but watch out for sums which may be in common.

1 and 4 have a HCF of 1

Find the highest common factor of the numeric parts of the product expressions.

x and x2 have x in common

Find the common factor for factors involving a variable.

cos2 x and cos x sin x have cos x in common.

Find the common factor for factors which involve trigonometric factors.

Hence xcos x is the largest common factor.


xcos2 x + 4x2 cos x sin x
=xcos x(cos x) + xcos x(4x sin x)
= xcos x(cos x + 4x sin x)

Put the parts together.


Try these.

Exercise 4

Factorise these then check your answers.

  1. sin 2x - cos 2x
  2. cos2 3x + cos 3x sin 2x
  3. (x - 1)2sin2 4y - sin 4y + xsin 4y
  4. tan2 z cos z - 3 sin z

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