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.
Expand $\sin\left(\theta+\frac{\pi}{4}\right)$sin(θ+π4).
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 $\sin\left(\theta+\frac{\pi}{4}\right)=\sin\theta\cos\frac{\pi}{4}+\sin\frac{\pi}{4}\cos\theta$sin(θ+π4)=sinθcosπ4+sinπ4cosθ
$=\frac{1}{\sqrt{2}}\left(\sin\theta+\cos\theta\right)$=1√2(sinθ+cosθ)
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)≡cosαcosβ−sinαsinβ, we have $\cos\left(x+2y\right)=\cos x\cos2y-\sin x\sin2y$cos(x+2y)=cosxcos2y−sinxsin2y.
Re-write this expression more concisely as a single trigonometric function: $\frac{\sqrt{3}}{2}\sin x-\frac{1}{2}\cos x$√32sinx−12cosx.
We recognise $\frac{\sqrt{3}}{2}$√32 as either $\sin\frac{\pi}{3}$sinπ3 or as $\cos\frac{\pi}{6}$cosπ6; and $\frac{1}{2}$12 as either $\sin\frac{\pi}{6}$sinπ6 or $\cos\frac{\pi}{3}$cosπ3. So, the original expression could be written as
$\cos\frac{\pi}{6}\sin x-\sin\frac{\pi}{6}\cos x$cosπ6sinx−sinπ6cosx.
This is, $\sin\left(x-\frac{\pi}{6}\right)$sin(x−π6).
Alternatively we could put
$\frac{\sqrt{3}}{2}\sin x-\frac{1}{2}\cos x=\sin\frac{\pi}{3}\sin x-\cos\frac{\pi}{3}\cos x$√32sinx−12cosx=sinπ3sinx−cosπ3cosx. This is,
$-\left(\cos\frac{\pi}{3}\cos x-\sin\frac{\pi}{3}\sin x\right)=-\cos\left(\frac{\pi}{3}+x\right)=\cos\left(\frac{\pi}{3}+x\right)$−(cosπ3cosx−sinπ3sinx)=−cos(π3+x)=cos(π3+x).
We can show that these two apparently different results are the same, by calling on facts about complementary functions.
Expand $\tan\left(\theta-x\right)$tan(θ−x).
Express $\sin A\cos2B+\cos A\sin2B$sinAcos2B+cosAsin2B using one trigonometric ratio.
Express $\cos\left(3\theta+x\right)\cos3\theta-\sin\left(3\theta+x\right)\sin3\theta$cos(3θ+x)cos3θ−sin(3θ+x)sin3θ in simplest form.