$π/7$ 三角函数求值

hbghlyj

$$8 = \csc\left(π\over7\right) \sec\left(3π\over14\right) \left(\csc\left(3π\over14\right) + \sec\left(π\over7\right)\right)$$

hbghlyj

WolframAlpha

$\text{Verify the following identity:}$
$${8}= \csc{{\left(\frac{\pi}{{7}}\right)}} \sec{{\left(\frac{{{3}\pi}}{{14}}\right)}}{\left( \csc{{\left(\frac{{{3}\pi}}{{14}}\right)}}+ \sec{{\left(\frac{\pi}{{7}}\right)}}\right)}$$
$\text{Write cosecant as 1/sine and secant as 1/cosine:}$
$${8}=\frac{1}{ \cos{{\left(\frac{{{3}\pi}}{{14}}\right)}}}{\left(\frac{1}{ \cos{{\left(\frac{\pi}{{7}}\right)}}}+\frac{1}{ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}}\right)}\frac{1}{ \sin{{\left(\frac{\pi}{{7}}\right)}}}$$
$\text{Put }\ \frac{1}{ \cos{{\left(\frac{\pi}{{7}}\right)}}}+\frac{1}{ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}}\ \text{ over the common denominator }\ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}} \cos{{\left(\frac{\pi}{{7}}\right)}}:\frac{1}{ \cos{{\left(\frac{\pi}{{7}}\right)}}}+\frac{1}{ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}}=\frac{{ \cos{{\left(\frac{\pi}{{7}}\right)}}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}}}{{ \cos{{\left(\frac{\pi}{{7}}\right)}} \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}}}:$
$${8}=\frac{1}{{ \cos{{\left(\frac{{{3}\pi}}{{14}}\right)}} \sin{{\left(\frac{\pi}{{7}}\right)}}}}{\left(\frac{{ \cos{{\left(\frac{\pi}{{7}}\right)}}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}}}{{ \cos{{\left(\frac{\pi}{{7}}\right)}} \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}}}\right)}$$
$\text{Multiply both sides by }\ \sin{{\left(\frac{\pi}{{7}}\right)}} \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}} \cos{{\left(\frac{\pi}{{7}}\right)}} \cos{{\left(\frac{{{3}\pi}}{{14}}\right)}}:$
$${8} \cos{{\left(\frac{\pi}{{7}}\right)}} \cos{{\left(\frac{{{3}\pi}}{{14}}\right)}} \sin{{\left(\frac{\pi}{{7}}\right)}} \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}= \cos{{\left(\frac{\pi}{{7}}\right)}}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}$$
$ \cos{{\left(\frac{\pi}{{7}}\right)}} \cos{{\left(\frac{{{3}\pi}}{{14}}\right)}}=\frac{1}{{2}}{\left( \cos{{\left(\frac{\pi}{{7}}-\frac{{{3}\pi}}{{14}}\right)}}+ \cos{{\left(\frac{\pi}{{7}}+\frac{{{3}\pi}}{{14}}\right)}}\right)}=\frac{1}{{2}}{\left( \cos{{\left(-\frac{\pi}{{14}}\right)}}+ \cos{{\left(\frac{{{5}\pi}}{{14}}\right)}}\right)}:$
$${\left({4} \sin{{\left(\frac{\pi}{{7}}\right)}} \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}{\left( \cos{{\left(-\frac{\pi}{{14}}\right)}}+ \cos{{\left(\frac{{{5}\pi}}{{14}}\right)}}\right)}\right)}= \cos{{\left(\frac{\pi}{{7}}\right)}}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}$$
Use the identity $\cos{{\left(-\frac{\pi}{{14}}\right)}}= \cos{{\left(\frac{\pi}{{14}}\right)}}$:
$${4} \sin{{\left(\frac{\pi}{{7}}\right)}} \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}{\left( \cos{{\left(\frac{\pi}{{14}}\right)}}+ \cos{{\left(\frac{{{5}\pi}}{{14}}\right)}}\right)}= \cos{{\left(\frac{\pi}{{7}}\right)}}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}$$
$ \sin{{\left(\frac{\pi}{{7}}\right)}} \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}=\frac{1}{{2}}{\left( \cos{{\left(\frac{\pi}{{7}}-\frac{{{3}\pi}}{{14}}\right)}}- \cos{{\left(\frac{\pi}{{7}}+\frac{{{3}\pi}}{{14}}\right)}}\right)}=\frac{1}{{2}}{\left( \cos{{\left(-\frac{\pi}{{14}}\right)}}- \cos{{\left(\frac{{{5}\pi}}{{14}}\right)}}\right)}:$
$${\left({2}{\left( \cos{{\left(-\frac{\pi}{{14}}\right)}}- \cos{{\left(\frac{{{5}\pi}}{{14}}\right)}}\right)}{\left( \cos{{\left(\frac{\pi}{{14}}\right)}}+ \cos{{\left(\frac{{{5}\pi}}{{14}}\right)}}\right)}\right)}= \cos{{\left(\frac{\pi}{{7}}\right)}}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}$$
$\text{Use the identity }\ \cos{{\left(-\frac{\pi}{{14}}\right)}}= \cos{{\left(\frac{\pi}{{14}}\right)}}:$
$${2}{\left( \cos{{\left(\frac{\pi}{{14}}\right)}}- \cos{{\left(\frac{{{5}\pi}}{{14}}\right)}}\right)}{\left( \cos{{\left(\frac{\pi}{{14}}\right)}}+ \cos{{\left(\frac{{{5}\pi}}{{14}}\right)}}\right)}= \cos{{\left(\frac{\pi}{{7}}\right)}}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}$$
${2}{\left( \cos{{\left(\frac{\pi}{{14}}\right)}}- \cos{{\left(\frac{{{5}\pi}}{{14}}\right)}}\right)}{\left( \cos{{\left(\frac{\pi}{{14}}\right)}}+ \cos{{\left(\frac{{{5}\pi}}{{14}}\right)}}\right)}={2}{ \cos{{\left(\frac{\pi}{{14}}\right)}}^{2}-}{2}{ \cos{{\left(\frac{{{5}\pi}}{{14}}\right)}}^{2}:}$
$${\left({2}{ \cos{{\left(\frac{\pi}{{14}}\right)}}^{2}-}{2} \cos{{\left(\frac{{{5}\pi}}{{14}}\right)}}^{2}\right)}= \cos{{\left(\frac{\pi}{{7}}\right)}}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}$$
${ \cos{{\left(\frac{\pi}{{14}}\right)}}^{2}=}\frac{1}{{2}}{\left( \cos{{\left(\frac{\pi}{{7}}\right)}}+{1}\right)}:$
$${2}\times{\left(\frac{{ \cos{{\left(\frac{\pi}{{7}}\right)}}+{1}}}{{2}}\right)}-{2}{ \cos{{\left(\frac{{{5}\pi}}{{14}}\right)}}^{2}=} \cos{{\left(\frac{\pi}{{7}}\right)}}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}$$
$\frac{{ \cos{{\left(\frac{\pi}{{7}}\right)}}+{1}}}{{2}}=\frac{1}{{2}} \cos{{\left(\frac{\pi}{{7}}\right)}}+\frac{1}{{2}}:$
$${2}{\left(\frac{1}{{2}}+\frac{{ \cos{{\left(\frac{\pi}{{7}}\right)}}}}{{2}}\right)}-{2}{ \cos{{\left(\frac{{{5}\pi}}{{14}}\right)}}^{2}=} \cos{{\left(\frac{\pi}{{7}}\right)}}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}$$
${ \cos{{\left(\frac{{{5}\pi}}{{14}}\right)}}^{2}=}\frac{1}{{2}}{\left({1}- \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}\right)}:$
$${2}{\left(\frac{{ \cos{{\left(\frac{\pi}{{7}}\right)}}}}{{2}}+\frac{1}{{2}}\right)}-{2}{\left(\frac{{{1}- \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}}}{{2}}\right)}= \cos{{\left(\frac{\pi}{{7}}\right)}}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}$$
$\frac{{{1}- \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}}}{{2}}=\frac{1}{{2}}-\frac{1}{{2}} \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}:$
$${2}{\left(\frac{{ \cos{{\left(\frac{\pi}{{7}}\right)}}}}{{2}}+\frac{1}{{2}}\right)}-{2}{\left(\frac{1}{{2}}-\frac{{ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}}}{{2}}\right)}= \cos{{\left(\frac{\pi}{{7}}\right)}}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}$$
${2}{\left(\frac{{ \cos{{\left(\frac{\pi}{{7}}\right)}}}}{{2}}+\frac{1}{{2}}\right)}= \cos{{\left(\frac{\pi}{{7}}\right)}}+{1}:$
$${\left( \cos{{\left(\frac{\pi}{{7}}\right)}}+{1}\right)}-{2}{\left(\frac{1}{{2}}-\frac{{ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}}}{{2}}\right)}= \cos{{\left(\frac{\pi}{{7}}\right)}}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}$$
$-{2}{\left(\frac{1}{{2}}-\frac{{ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}}}{{2}}\right)}= \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}-{1}:$
$${1}+ \cos{{\left(\frac{\pi}{{7}}\right)}}+{\left( \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}-{1}\right)}= \cos{{\left(\frac{\pi}{{7}}\right)}}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}$$
${1}+ \cos{{\left(\frac{\pi}{{7}}\right)}}-{1}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}= \cos{{\left(\frac{\pi}{{7}}\right)}}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}:$
$${\left( \cos{{\left(\frac{\pi}{{7}}\right)}}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}\right)}= \cos{{\left(\frac{\pi}{{7}}\right)}}+ \sin{{\left(\frac{{{3}\pi}}{{14}}\right)}}$$

kuing

想起了这帖 forum.php?mod=viewthread&tid=6809 不知有否关联🤔

exctttt

$8\sin\dfrac{\pi}{7}\cos\dfrac{\pi}{7}\sin\dfrac{3\pi}{14}\cos\dfrac{3\pi}{14}=\cos\dfrac{\pi}{7}+\sin\dfrac{3\pi}{14}$ 之后
左边二倍角
$2\sin\dfrac{2\pi}{7}\sin\dfrac{3\pi}{7}=\cos\dfrac{\pi}{7}+\sin\dfrac{3\pi}{14}$
积化和差
$\cos\dfrac{\pi}{7}-\cos\dfrac{5\pi}{7}=\cos\dfrac{\pi}{7}+\sin\dfrac{3\pi}{14}$
显然成立