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Question

# Express the Law of Sines in terms of only the cosine function.

Solution

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According to law of sines:

$\dfrac{a}{\sin a}=\dfrac{b}{\sin b}=\dfrac{c}{\sin c}$

We know $\sin (90 \text{\textdegree}+\theta)=\cos \theta$.

Expressing law of sines in terms of cosines:

\begin{align*} \dfrac{a}{\sin a}&=\dfrac{b}{\sin b}=\dfrac{c}{\sin c}\\ \dfrac{a}{\sin (90\text{\textdegree} -90\text{\textdegree} + a)}&=\dfrac{b}{\sin(90\text{\textdegree} -90\text{\textdegree} + b)}=\dfrac{c}{\sin(90\text{\textdegree} -90\text{\textdegree} + c)}\\ \textcolor{#4257b2}{\dfrac{a}{\cos (a-90\text{\textdegree} )}}&=\textcolor{#4257b2}{\dfrac{b}{\cos( b-90\text{\textdegree} )}=\dfrac{c}{\cos( c-90\text{\textdegree} )}}\\ \end{align*}

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