General solution to $\mathrm{sin}\alpha +\mathrm{sin}\beta $ and $\mathrm{cos}\alpha +\mathrm{cos}\beta $?

rjawbreakerca
2022-07-07
Answered

General solution to $\mathrm{sin}\alpha +\mathrm{sin}\beta $ and $\mathrm{cos}\alpha +\mathrm{cos}\beta $?

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asked 2021-08-20

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

asked 2022-07-07

the following equality is given:

$2\sqrt{2}\mathrm{sin}x+\sqrt{2}\mathrm{cos}x=\sqrt{-\mathrm{sin}2x}$

$2\sqrt{2}\mathrm{sin}x+\sqrt{2}\mathrm{cos}x=\sqrt{-\mathrm{sin}2x}$

asked 2022-05-29

Solve, in the interval ${0}^{\circ}\le \theta \le {360}^{\circ}$

$\mathrm{cos}(\theta +{25}^{\circ})+\mathrm{sin}(\theta +{65}^{\circ})=1$

$\mathrm{cos}(\theta +{25}^{\circ})+\mathrm{sin}(\theta +{65}^{\circ})=1$

asked 2022-06-23

What is the value of the expression $\mathrm{sin}\frac{2\pi}{7}\mathrm{sin}\frac{4\pi}{7}+\mathrm{sin}\frac{4\pi}{7}\mathrm{sin}\frac{8\pi}{7}+\mathrm{sin}\frac{8\pi}{7}\mathrm{sin}\frac{2\pi}{7}$

asked 2022-01-23

Proving that $\mathrm{sec}\frac{\pi}{30}=\sqrt{2-\sqrt{5}+\sqrt{15-6\sqrt{5}}}$

which I instantly wanted to prove.

I know that I can "reduce" the problem to the evaluation of$\mathrm{cos}\frac{\pi}{15}$ , as the rest is easy with the use of the half-angle formula.

I know that cos obeys the nice relation

$\mathrm{cos}nx={T}_{n}\left(\mathrm{cos}x\right)$

where

$T}_{n}\left(x\right)=\frac{n}{2}\sum _{k=0}^{\left|\frac{n}{2}\right|}\frac{{(-1)}^{k}}{n-k}\left(\begin{array}{c}n-k\\ k\end{array}\right){\left(2x\right)}^{n-2k$

Thus, setting$t=\mathrm{cos}\frac{\pi}{15}$ ,

${T}_{15}\left(t\right)=-1$

The only thing left to do is solve for t. We can narrow down our search to the values 0<t<1.

I have never dealt with degree-15 polynomials before, so I was hoping one of you could help me out.

which I instantly wanted to prove.

I know that I can "reduce" the problem to the evaluation of

I know that cos obeys the nice relation

where

Thus, setting

The only thing left to do is solve for t. We can narrow down our search to the values 0<t<1.

I have never dealt with degree-15 polynomials before, so I was hoping one of you could help me out.

asked 2022-05-21

Expand $(\mathrm{tan}x-\sqrt{3})(3\mathrm{tan}x+\sqrt{3})$

asked 2022-02-26

Which is the easiest way to evaluate:

${\int}_{0}^{\frac{\pi}{2}}(\sqrt{\mathrm{tan}x}+\sqrt{\mathrm{cot}x})$