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The law of sines and law of cosines can be used to find angle and sides measures in any triangle?

Analytic geometry
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Analytic geometry
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Formulas: sinAcot(productb)=sinBcot(producta), cosAcos(productb)cos(productc)...

Analytic geometry
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pablos28spainzd
2022-07-15

$AB\equiv {a}_{1}x+{b}_{1}y+{c}_{1}=0\phantom{\rule{0ex}{0ex}}AC\equiv {a}_{2}x+{b}_{2}y+{c}_{2}=0\phantom{\rule{0ex}{0ex}}BC\equiv px+qy+r=0$

Then an angle $\mathrm{\angle}A$ is acute if

$\left|\begin{array}{cc}{a}_{1}& {b}_{1}\\ p& q\end{array}\right|\left|\begin{array}{cc}p& q\\ {a}_{2}& {b}_{2}\end{array}\right|({a}_{1}{a}_{2}+{b}_{1}{b}_{2})<0$

And the angle $\mathrm{\angle}A$ is obtuse if

$\left|\begin{array}{cc}{a}_{1}& {b}_{1}\\ p& q\end{array}\right|\left|\begin{array}{cc}p& q\\ {a}_{2}& {b}_{2}\end{array}\right|({a}_{1}{a}_{2}+{b}_{1}{b}_{2})>0$

But I absolutely have no idea how to prove this theorem.

Analytic geometry
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Caleb Proctor
2022-07-12

Let E and F be the midpoints of (AC) and (BD), $EF\cap AB=\{J\}$ , $EF\cap CD=\{K\}$ , $CJ\cap BK=\{L\}$ . If M is the midpoint of (KJ), show that O-M-L are collinear.

Analytic geometry
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lilmoore11p8
2022-07-10

Analytic geometry
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Janet Forbes
2022-07-10

$F(x,y,z)={x}^{2}+{y}^{2}-{z}^{2}=1$

at

$({x}_{0},{y}_{0},{z}_{0})$

The equation of the tangent plane is

${x}_{0}x+{y}_{0}y-{z}_{0}z=1$

Analytic geometry
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Kolten Conrad
2022-07-08

I have three vectors, $\overrightarrow{a},\overrightarrow{b},$ and $\overrightarrow{c}$ in n-dimensional space. I know the coordinates of all three vectors and their dot products. Both $\overrightarrow{a}$ and $\overrightarrow{b}$ are rotated away from $\overrightarrow{c}$ by an angle $\alpha $, in their own respective directions, obtaining ${\overrightarrow{a}}^{\prime}$ and ${\overrightarrow{b}}^{\prime}$ . What is the angle between ${\overrightarrow{a}}^{\prime}$ and ${\overrightarrow{b}}^{\prime}$ ?

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