Blog: Express sin32+sin54 as sum or difference.

Hello!

This expression is already a sum of two numbers, $\displaystyle{\sin{{\left({32}\right)}}}$ and $\displaystyle{\sin{{\left({54}\right)}}}.$ Probably you want or express it as a product, or as an expression involving trigonometric functions of sum or difference of the arguments, $\displaystyle{32}$ and $\displaystyle{54}.$

For this, we need the well-known formula

$\displaystyle{\sin{{\left({x}\right)}}}+{\sin{{\left({y}\right)}}}={2}{\sin{{\left(\frac{{{x}+{y}}}{{2}}\right)}}}{\cos{{\left(\frac{{{x}-{y}}}{{2}}\right)}}},$

which is true for any real $\displaystyle{x}$ and $\displaystyle{y}.$

For the given numbers it gives us

$\displaystyle{\sin{{\left({32}\right)}}}+{\sin{{\left({54}\right)}}}={2}{\sin{{\left(\frac{{{32}+{54}}}{{2}}\right)}}}{\cos{{\left(\frac{{{32}-{54}}}{{2}}\right)}}}=$

$\displaystyle={2}{\sin{{\left({43}\right)}}}{\cos{{\left(-{11}\right)}}}\ldots$

Hello!

For this, we need the well-known formula

$\displaystyle{\sin{{\left({x}\right)}}}+{\sin{{\left({y}\right)}}}={2}{\sin{{\left(\frac{{{x}+{y}}}{{2}}\right)}}}{\cos{{\left(\frac{{{x}-{y}}}{{2}}\right)}}},$

which is true for any real $\displaystyle{x}$ and $\displaystyle{y}.$

For the given numbers it gives us

$\displaystyle{\sin{{\left({32}\right)}}}+{\sin{{\left({54}\right)}}}={2}{\sin{{\left(\frac{{{32}+{54}}}{{2}}\right)}}}{\cos{{\left(\frac{{{32}-{54}}}{{2}}\right)}}}=$

$\displaystyle={2}{\sin{{\left({43}\right)}}}{\cos{{\left(-{11}\right)}}}={2}{\sin{{\left({43}\right)}}}{\cos{{\left({11}\right)}}}.$

If you misspelled the expression, and it is actually $\displaystyle{\sin{{\left({32}\right)}}}\cdot{\sin{{\left({54}\right)}}},$ then we really can express it as a sum with the help of the formula

$\displaystyle{\sin{{\left({x}\right)}}}{\sin{{\left({y}\right)}}}=\frac{{1}}{{2}}{\left({\cos{{\left({x}-{y}\right)}}}-{\cos{{\left({x}+{y}\right)}}}\right)},$

for the given numbers it is

$\displaystyle{\sin{{\left({32}\right)}}}{\sin{{\left({54}\right)}}}=\frac{{1}}{{2}}{\left({\cos{{\left(-{22}\right)}}}-{\cos{{\left({86}\right)}}}\right)}=\frac{{1}}{{2}}{\left({\cos{{\left({22}\right)}}}-{\cos{{\left({86}\right)}}}\right)}.$