# Simplify $\sin{(\theta+60^°)}$ $-$ $\cos{(\theta+30^°)}$

$\sin{(\theta+60^°)}$ $-$ $\cos{(\theta+30^°)}$ is a trigonometric expression and its value can be evaluated by simplifying it. The both trigonometric functions have angles which are sum of two angles so that the value of the trigonometric expression can be evaluated by using angle sum identities.

### Expand sine of sum of angles

The trigonometric term $\sin{(\theta+60^°)}$ can be expanded by using sin of sum of two angles trigonometric identity.

$\sin{(\theta+60^°)}$ $\,=\,$ $\sin{\theta}\cos{(60^°)}$ $+$ $\sin{\theta}\cos{(60^°)}$

$\implies \sin{(\theta+60^°)}$ $\,=\,$ $\sin{\theta} \times \cos{(60^°)}$ $+$ $\cos{\theta} \times \sin{(60^°)}$

$\implies \sin{(\theta+60^°)}$ $\,=\,$ $\sin{\theta} \times \dfrac{1}{2}$ $+$ $\cos{\theta} \times \dfrac{\sqrt{3}}{2}$

$\,\,\, \therefore \,\,\,\,\,\, \sin{(\theta+60^°)}$ $\,=\,$ $\dfrac{\sin{\theta}}{2}$ $+$ $\dfrac{\sqrt{3}\cos{\theta}}{2}$

### Expand cosine of sum of angles

Similarly, the trigonometric term $\cos{(\theta+30^°)}$ can also be expanded by using cos of sum of two angles trigonometric identity.

$\cos{(\theta+30^°)}$ $\,=\,$ $\cos{\theta}\cos{(30^°)}$ $-$ $\sin{\theta}\sin{(30^°)}$

$\implies \cos{(\theta+30^°)}$ $\,=\,$ $\cos{\theta} \times \cos{(30^°)}$ $-$ $\sin{\theta} \times \sin{(30^°)}$

$\implies \cos{(\theta+30^°)}$ $\,=\,$ $\cos{\theta} \times \dfrac{\sqrt{3}}{2}$ $-$ $\sin{\theta} \times \dfrac{1}{2}$

$\,\,\, \therefore \,\,\,\,\,\, \cos{(\theta+30^°)}$ $\,=\,$ $\dfrac{\sqrt{3}\cos{\theta}}{2}$ $-$ $\dfrac{\sin{\theta}}{2}$

### Evaluate Trigonometric Expression

Finally, subtract $\cos{(\theta+30^°)}$ from $\sin{(\theta+60^°)}$ to get the value of the trigonometric expression mathematically.

$\sin{(\theta+60^°)}$ $-$ $\cos{(\theta+30^°)}$ $\,=\,$ $\Bigg(\dfrac{\sin{\theta}}{2}$ $+$ $\dfrac{\sqrt{3}\cos{\theta}}{2}\Bigg)$ $-$ $\Bigg(\dfrac{\sqrt{3}\cos{\theta}}{2}$ $-$ $\dfrac{\sin{\theta}}{2}\Bigg)$

$\implies \sin{(\theta+60^°)}$ $-$ $\cos{(\theta+30^°)}$ $\,=\,$ $\dfrac{\sin{\theta}}{2}$ $+$ $\dfrac{\sqrt{3}\cos{\theta}}{2}$ $-$ $\dfrac{\sqrt{3}\cos{\theta}}{2}$ $+$ $\dfrac{\sin{\theta}}{2}$

$\implies \sin{(\theta+60^°)}$ $-$ $\cos{(\theta+30^°)}$ $\,=\,$ $\dfrac{\sin{\theta}}{2}$ $+$ $\require{\cancel} \cancel{\dfrac{\sqrt{3}\cos{\theta}}{2}}$ $-$ $\require{\cancel} \cancel{\dfrac{\sqrt{3}\cos{\theta}}{2}}$ $+$ $\dfrac{\sin{\theta}}{2}$

$\implies \sin{(\theta+60^°)}$ $-$ $\cos{(\theta+30^°)}$ $\,=\,$ $\dfrac{\sin{\theta}}{2}$ $+$ $\dfrac{\sin{\theta}}{2}$

$\implies \sin{(\theta+60^°)}$ $-$ $\cos{(\theta+30^°)}$ $\,=\,$ $\dfrac{2\sin{\theta}}{2}$

$\implies \sin{(\theta+60^°)}$ $-$ $\cos{(\theta+30^°)}$ $\,=\,$ $\require{cancel} \dfrac{\cancel{2} \sin{\theta}}{\cancel{2}}$

$\,\,\, \therefore \,\,\,\,\,\, \sin{(\theta+60^°)}$ $-$ $\cos{(\theta+30^°)}$ $\,=\,$ $\sin{\theta}$

A best free mathematics education website that helps students, teachers and researchers.

###### Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

###### Maths Problems

A math help place with list of solved problems with answers and worksheets on every concept for your practice.

Learn solutions

###### Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.