e1s2kat26

2021-08-11

Express the product $\mathrm{sin}6x×\mathrm{sin}4x$ as a sum or difference.

SchepperJ

Given:
To express the product $\mathrm{sin}6x×\mathrm{sin}4x$ as sum or express it as difference.
Concept:
The study of side lengths and angles of triangle is studied by a relation called trigonometry.
The equation given is related to the expression below,
$\mathrm{sin}\left(6x\right)×\mathrm{sin}\left(4x\right)=\frac{-1}{2}\left(-2×\mathrm{sin}\left(6x\right)×\mathrm{sin}\left(4\right)\right)$
To obtain the relation derivate the right hand side by sin and cos rule,
$=\frac{-1}{2}\left(\mathrm{cos}\left(6x+4x\right)-\mathrm{cos}\left(6x-4x\right)\right)$
The values of cos are sum and differenced to obtain the relation below,
$=\frac{-1}{2}\left(\mathrm{cos}\left(10x\right)-\mathrm{cos}\left(2x\right)\right)$
Thus the product equation is expressed as sum or difference as below,
$\mathrm{sin}\left(6x\right)×\mathrm{sin}\left(4x\right)=\frac{-1}{2}\left(\mathrm{cos}\left(10x\right)-\mathrm{cos}\left(2x\right)\right)$

Jeffrey Jordon