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India
Class XI

Expand expressions using double and half angle identities

Lesson

Where the argument of a trigonometric function is a multiple of some variable, say $2x$2x or $3x$3x or $\frac{x}{2}$x2, we may wish to express the function in terms of the variable $x$x itself. In other situations, we may wish to work in the opposite direction write a complicated expression more simply in terms of a multiple argument.

Example 1

The expression $\cos3x$cos3x can be expressed in terms of $\cos x$cosx.

One way to do this is to think of the argument $3x$3x in the form $2x+x$2x+x. Then, $\cos3x=\cos\left(2x+x\right)$cos3x=cos(2x+x)$=\cos2x\cos x-\sin2x\sin x$=cos2xcosxsin2xsinx

We further expand $\cos2x$cos2x and $\sin2x$sin2x to obtain

$\cos3x=\left(\cos^2x-\sin^2x\right)\cos x-2\sin^2x\cos x$cos3x=(cos2xsin2x)cosx2sin2xcosx

Because the required end result is to be in terms of the cosine function, we must replace $\sin^2x$sin2x with $1-\cos^2x$1cos2x.

Then,

$\cos3x=\left(2\cos^2x-1\right)\cos x-2\left(1-\cos^2x\right)\cos x$cos3x=(2cos2x1)cosx2(1cos2x)cosx and finally, on collecting like terms, we find $\cos3x\equiv4\cos^3x-3\cos x$cos3x4cos3x3cosx

 

Example 2

The expression $\sin^4x-\cos^4x$sin4xcos4x can be written more simply in terms of a multiple angle.

The expression is the difference of two squares.

It can be factorised to $\left(\sin^2x+\cos^2x\right)\left(\sin^2x-\cos^2x\right)$(sin2x+cos2x)(sin2xcos2x).

The first bracket is just $1$1, and the second bracket is the negative of the expansion of $\cos2x$cos2x.

So, we have $\sin^4x-\cos^4x\equiv-\cos2x$sin4xcos4xcos2x

 

Example 3

It is possible to deduce exact values for the trigonometric functions of angles that are multiples of $\frac{\pi}{10}$π10 or $18^\circ$18°. Such angles are related to angles found in the regular pentagon.

Suppose we look for an angle $\theta$θ such that $\cos3\theta=\sin2\theta$cos3θ=sin2θ. We found in Example 1 above, that $\cos3\theta=4\cos^3\theta-3\cos\theta$cos3θ=4cos3θ3cosθ. We also know that $\sin2\theta=2\sin\theta\cos\theta$sin2θ=2sinθcosθ. Thus, we can write

$4\cos^3\theta-3\cos\theta=2\sin\theta\cos\theta$4cos3θ3cosθ=2sinθcosθ. On cancelling the common factor $\cos\theta$cosθ from each term and replacing $\cos^2\theta$cos2θ with $1-\sin^2\theta$1sin2θ, we have a quadratic in $\sin\theta$sinθ.

$4\sin^2\theta+2\sin\theta-1=0$4sin2θ+2sinθ1=0

This has a positive solution $\sin\theta=\frac{1}{4}\left(\sqrt{5}-1\right)$sinθ=14(51)

But what is the angle $\theta$θ? We began with the equation $\cos3\theta=\sin2\theta$cos3θ=sin2θ. In the first quadrant, where $\sin$sin and $\cos$cos are both positive,  $\sin x\equiv\cos\left(\frac{\pi}{2}-x\right)$sinxcos(π2x). So, 

$\cos3\theta=\cos\left(\frac{\pi}{2}-2\theta\right)$cos3θ=cos(π22θ). It must be the case that $3\theta=\frac{\pi}{2}-2\theta$3θ=π22θ and hence, 

$\theta=\frac{\pi}{10}$θ=π10

Thus,

$\sin\frac{\pi}{10}=\frac{1}{4}\left(\sqrt{5}-1\right)$sinπ10=14(51)

With the Pythagorean identity and with the definition of the tangent function, we can deduce that

$\cos\frac{\pi}{10}=\frac{1}{4}\sqrt{10+2\sqrt{5}}$cosπ10=1410+25

and

$\tan\frac{\pi}{10}=\frac{\sqrt{5}-1}{\sqrt{10+2\sqrt{5}}}$tanπ10=5110+25

Worked Examples

QUESTION 1

Fully simplify $\sin x\cos^3\left(x\right)-\sin^3\left(x\right)\cos x$sinxcos3(x)sin3(x)cosx, using trigonometric identities where necessary.

QUESTION 2

Write $\sin3x$sin3x in terms of $\sin x$sinx.

  1. Leave your answer in expanded form.

QUESTION 3

Using a double angle identity, simplify the expression $\sin5x\cos5x$sin5xcos5x.

Express your answer as a single trigonometric function.

Outcomes

11.SF.TF.1

Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity sin^2 x + cos^2 x = 1, for all x. Signs of trigonometric functions and sketch of their graphs. Expressing sin (x + y) and cos (x + y) in terms of sin x, sin y, cos x and cos y. Deducing the identities like following: cot(x + or - y), sin x + sin y, cos x + cos y, sin x - sin y, cos x - cos y (see syllabus document)

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