345: 2 x 2 Special Orthogonal Matrix Can Be Expressed with Sine and Cosine of Angle

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A description/proof of that 2 x 2 special orthogonal matrix can be expressed with sine and cosine of angle

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Topics

About: matrix

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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Starting Context

• The reader knows a definition of matrix.

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Target Context

• The reader will have a description and a proof of the proposition that any 2 x 2 special orthogonal matrix can be expressed with the sine and the cosine of an angle.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Description

Any 2 x 2 special orthogonal matrix, $M$, can be expressed as $\left(\begin{array}{cc}sin\theta & -cos\theta \\ cos\theta & sin\theta \end{array}\right)$ where $\theta$ is an angle such that $0\le \theta <2\pi$.

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2: Proof

Let $M$ be $\left(\begin{array}{cc}a& c\\ b& d\end{array}\right)$. $M^{t}M=\left(\begin{array}{cc}{a}^2+b^2& ac+bd\\ ac+bd& c^2+d^2\end{array}\right)=I$ and $detM=ad-bc=1$. $a=sin\theta ,b=cos\theta$ for a $\theta$ such that $0\le \theta <2\pi$. $c=-cos{\theta }^{\prime },d=sin{\theta }^{\prime }$ for a ${\theta }^{\prime }$ such that $0\le {\theta }^{\prime }<2\pi$. $-sin\theta cos{\theta }^{\prime }+cos\theta sin{\theta }^{\prime }=0$, $tan\theta =tan{\theta }^{\prime }$. ${\theta }^{\prime }=\theta$ or ${\theta }^{\prime }=\theta +\pi$ or ${\theta }^{\prime }=\theta -\pi$. $sin\theta sin{\theta }^{\prime }+cos\theta cos{\theta }^{\prime }=1$. So, ${\theta }^{\prime }=\theta$.

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References

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