write the product as a sum.\n19 sin(\\(\\frac{x}{2}\\)) cos(\\(\\frac{x}{4}\\))
write the product as a sum.\n19 sin(\\(\\frac{x}{2}\\)) cos(\\(\\frac{x}{4}\\))
Answer
Explanation:
Step1: Use the product - to - sum formula
The formula for ( \sin A\cos B=\frac{1}{2}[\sin(A + B)+\sin(A - B)]). Here (A=\frac{x}{2}) and (B = \frac{x}{4}). So, (19\sin(\frac{x}{2})\cos(\frac{x}{4})=19\times\frac{1}{2}[\sin(\frac{x}{2}+\frac{x}{4})+\sin(\frac{x}{2}-\frac{x}{4})])
Step2: Simplify the arguments of the sine functions
Simplify (\frac{x}{2}+\frac{x}{4}=\frac{2x + x}{4}=\frac{3x}{4}) and (\frac{x}{2}-\frac{x}{4}=\frac{2x-x}{4}=\frac{x}{4}) Then (19\times\frac{1}{2}[\sin(\frac{3x}{4})+\sin(\frac{x}{4})]=\frac{19}{2}\sin(\frac{3x}{4})+\frac{19}{2}\sin(\frac{x}{4}))
Answer:
(\frac{19}{2}\sin(\frac{3x}{4})+\frac{19}{2}\sin(\frac{x}{4}))