Another way to express $\text{cis}{75}^{\circ}+\text{cis}{83}^{\circ}+\text{cis}{91}^{\circ}+\cdots +\text{cis}{147}^{\circ}$

Hailie Blevins
2022-06-25
Answered

Another way to express $\text{cis}{75}^{\circ}+\text{cis}{83}^{\circ}+\text{cis}{91}^{\circ}+\cdots +\text{cis}{147}^{\circ}$

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Let P(x, y) be the terminal point on the unit circle determined by t. Then

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I have $\frac{{e}^{yi}-{e}^{-yi}}{{e}^{yi}+{e}^{-yi}}$ , and I know that this equals

$\frac{\frac{1}{2}({e}^{yi}-{e}^{-yi})}{\frac{1}{2}({e}^{yi}+{e}^{-yi})}=\frac{\text{sinh}\left\{yi\right\}}{\text{cosh}\left\{yi\right\}}=\text{tanh}\left\{yi\right\}$

However, this is also supposed to equal$i\mathrm{tan}y$ . I'm not sure how to get to this result.

I tried doing

$\frac{{e}^{yi}-{e}^{-yi}}{{e}^{yi}+{e}^{-yi}}=\frac{\frac{{e}^{2yi}-1}{{e}^{yi}}}{\frac{{e}^{2yi}+1}{{e}^{yi}}}=\frac{{e}^{2yi}-1}{{e}^{2yi}+1}$

However, this is also supposed to equal

I tried doing

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Show that in a $\mathrm{\Delta}ABC$, $\mathrm{sin}\frac{A}{2}\le \frac{a}{b+c}$

Hence or otherwise show that ${\mathrm{csc}}^{n}\frac{A}{2}+{\mathrm{csc}}^{n}\frac{B}{2}+{\mathrm{csc}}^{n}\frac{C}{2}$ has the minimum value ${3.2}^{n}$ for all $n\ge 1$

Hence or otherwise show that ${\mathrm{csc}}^{n}\frac{A}{2}+{\mathrm{csc}}^{n}\frac{B}{2}+{\mathrm{csc}}^{n}\frac{C}{2}$ has the minimum value ${3.2}^{n}$ for all $n\ge 1$

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How to solve $\mathrm{cos}\left({\mathrm{sin}}^{-1}(-\frac{3}{5})\right)$ ?

I tried solving it using the formula${\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x=1$ and I also got $\frac{45}{}$ as the answer but I got it by inputting the value of cosx in the given question in place of

${\mathrm{sin}}^{-1}(-\frac{3}{5})$

But I cant see any logic there.

I tried solving it using the formula

But I cant see any logic there.