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−  This unit looks at trigonometric formulae known as the double angle formulae. They are called
 
−  this because they involve trigonometric functions of double angles, i.e. sin 2A, cos 2A and tan2A.
 
−  In order to master the techniques explained here it is vital that you undertake the practice
 
−  exercises provided.
 
   
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−  After reading this text, and/or viewing the video tutorial on this topic, you should be able to:
 
−  • derive the double angle formulae from the addition formulae
 
−  • write the formula for cos 2A in alternative forms
 
−  • use the formulae to write trigonometric expressions in different forms
 
−  • use the formulae in the solution of trigonometric equations
 
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−  Contents
 
−  1. Introduction 2
 
−  2. The double angle formulae for sin 2A, cos 2A and tan2A 2
 
−  3. The formula cos2A = cos2 A − sin2 A 3
 
−  4. Finding sin3x in terms of sin x 3
 
−  5. Using the formulae to solve an equation 4
 
−  1. Introduction
 
−  This unit looks at trigonometric formulae known as the double angle formulae. They are called
 
−  this because they involve trigonometric functions of double angles, i.e. sin 2A, cos 2A and tan2A.
 
−  2. The double angle formulae for sin 2A, cos 2A and tan 2A
 
−  We start by recalling the addition formulae which have already been described in the unit of the
 
−  same name.
 
−  sin(A + B) = sin A cos B + cos A sin B
 
−  cos(A + B) = cos A cos B − sin A sin B
 
−  tan(A + B) = tan A + tan B
 
−  1 − tan A tan B
 
−  We consider what happens if we let B equal to A. Then the first of these formulae becomes:
 
−  sin(A + A) = sin A cos A + cos A sin A
 
−  so that
 
−  sin 2A = 2 sin A cos A
 
−  This is our first doubleangle formula, so called because we are doubling the angle (as in 2A).
 
−  Similarly, if we put B equal to A in the second addition formula we have
 
−  cos(A + A) = cos A cos A − sin A sin A
 
−  so that
 
−  cos 2A = cos2 A − sin2 A
 
−  and this is our second double angle formula.
 
−  Similarly
 
−  tan(A + A) = tan A + tan A
 
−  1 − tan A tan A
 
−  so that
 
−  tan 2A =
 
−  2 tan A
 
−  1 − tan2 A
 
−  These three double angle formulae should be learnt.
 
−  3. The formula cos 2A = cos2 A − sin2 A
 
−  We now examine this formula more closely.
 
−  We know from an important trigonometric identity that
 
−  cos2 A + sin2 A = 1
 
−  so that by rearrangement
 
−  sin2 A = 1 − cos2 A.
 
−  So using this result we can replace the term sin2 A in the double angle formula. This gives
 
−  cos 2A = cos2 A − sin2 A
 
−  = cos2 A − (1 − cos2 A)
 
−  = 2 cos2 A − 1
 
−  This is another double angle formula for cos 2A.
 
−  Alternatively we could replace the term cos2 A by 1 − sin2 A which gives rise to:
 
−  cos 2A = cos2 A − sin2 A
 
−  = (1 − sin2 A) − sin2 A
 
−  = 1 − 2 sin2 A
 
−  which is yet a third form.
 
−  4. Finding sin 3x in terms of sin x
 
−  Example
 
−  Consider the expression sin 3x. We will use the addition formulae and double angle formulae to
 
−  write this in a different form using only terms involving sin x and its powers.
 
−  We begin by thinking of 3x as 2x + x and then using an addition formula:
 
−  sin 3x = sin(2x + x)
 
−  = sin 2x cos x + cos 2x sin x using the first addition formula
 
−  = (2 sin x cos x) cos x + (1 − 2 sin2 x) sin x using the double angle formula
 
−  cos 2x = 1 − 2 sin2 x
 
−  = 2 sin x cos2 x + sin x − 2 sin3 x
 
−  = 2 sin x(1 − sin2 x) + sin x − 2 sin3 x from the identity cos2 x + sin2 x = 1
 
−  = 2 sin x − 2 sin3 x + sin x − 2 sin3 x
 
−  = 3 sin x − 4 sin3 x
 
−  We have derived another identity
 
−  sin 3x = 3 sin x − 4 sin3 x
 
−  Note that by using these formulae we have written sin 3x in terms of sin x (and its powers). You
 
−  could carry out a similar exercise to write cos 3x in terms of cos x.
 
−  5. Using the formulae to solve an equation
 
−  Example
 
−  Suppose we wish to solve the equation cos 2x = sin x, for values of x in the interval −π ≤ x < π.
 
−  We would like to try to write this equation so that it involves just one trigonometric function, in
 
−  this case sin x. To do this we will use the double angle formula
 
−  cos 2x = 1 − 2 sin2 x
 
−  The given equation becomes
 
−  1 − 2 sin2
 
−  x = sin x
 
−  which can be rewritten as
 
−  0 = 2 sin2 x + sin x − 1
 
−  This is a quadratic equation in the variable sin x. It factorises as follows:
 
−  0 = (2 sin x − 1)(sin x + 1)
 
−  It follows that one or both of these brackets must be zero:
 
−  2 sin x − 1 = 0 or sin x + 1 = 0
 
−  so that
 
−  sin x =
 
−  1
 
−  2
 
−  or sin x = −1
 