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Grade 12

Expand expressions using angle sum and difference identities

Lesson

In the chapter on Angle Sum and Difference Identities, we showed how identities involving a trigonometric function of a sum or difference have been derived. 

Here we show some examples of their use. The identities can be used to write an expression in expanded form. That is, avoiding functions of a sum or difference. It is also sometimes useful to recognise when it may be possible to perform this process in reverse.

Example 1

Expand $\sin\left(\theta+45^\circ\right)$sin(θ+45°).

We use the identity: $\sin\left(\alpha+\beta\right)\equiv\sin\alpha\cos\beta+\sin\beta\cos\alpha$sin(α+β)sinαcosβ+sinβcosα. It follows that $\sin\left(\theta+45^\circ\right)=\sin\theta\cos45^\circ+\sin45^\circ\cos\theta$sin(θ+45°)=sinθcos45°+sin45°cosθ

$=\frac{1}{\sqrt{2}}\left(\sin\theta+\cos\theta\right)$=12(sinθ+cosθ)

Example 2

Rewrite $\cos\left(x+2y\right)$cos(x+2y) using functions of $x$x and $y$y only.

Using the identity $\cos\left(\alpha+\beta\right)\equiv\cos\alpha\cos\beta-\sin\alpha\sin\beta$cos(α+β)cosαcosβsinαsinβ, we have $\cos\left(x+2y\right)=\cos x\cos2y-\sin x\sin2y$cos(x+2y)=cosxcos2ysinxsin2y.

Example 3

Re-write this expression more concisely as a single trigonometric function: $\frac{\sqrt{3}}{2}\sin x-\frac{1}{2}\cos x$32sinx12cosx.

We recognise $\frac{\sqrt{3}}{2}$32 as either $\sin60^\circ$sin60° or as $\cos30^\circ$cos30°; and $\frac{1}{2}$12 as either $\sin30^\circ$sin30° or $\cos60^\circ$cos60°. So, the original expression could be written as 

$\cos30^\circ\sin x-\sin30^\circ\cos x$cos30°sinxsin30°cosx.

This is,  $\sin\left(x-30^\circ\right)$sin(x30°).

Alternatively we could put 

$\frac{\sqrt{3}}{2}\sin x-\frac{1}{2}\cos x=\sin60^\circ\sin x-\cos60^\circ\cos x$32sinx12cosx=sin60°sinxcos60°cosx. This is,

$-\left(\cos60^\circ\cos x-\sin60^\circ\sin x\right)=-\cos\left(60^\circ+x\right)=\cos\left(60^\circ+x\right)$(cos60°cosxsin60°sinx)=cos(60°+x)=cos(60°+x).

We can show that these two apparently different results are the same, by calling on facts about complementary functions.

More Worked Examples

QUESTION 1

Express $\sin A\cos2B+\cos A\sin2B$sinAcos2B+cosAsin2B using one trigonometric ratio.

QUESTION 2

Expand $\sin\left(3x-y\right)$sin(3xy), rewriting it in terms of angles $3x$3x and $y$y.

QUESTION 3

Express $\cos\left(3\theta+x\right)\cos3\theta-\sin\left(3\theta+x\right)\sin3\theta$cos(3θ+x)cos3θsin(3θ+x)sin3θ in simplest form.

Outcomes

12F.B.3.2

Explore the algebraic development of the compound angle formulas, and use the formulas to determine exact values of trigonometric ratios

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