Law of Sines

The Law of Sines helps us solve for non-right triangles. It is simply a proportion that we need to solve for.

    \[\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}\]

or

    \[\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}\]

 

Capital letters always represent angles.
Lowercase letters always represent sides.

Angle A is always opposite side a.
Angle B is always opposite side b.
Angle C is always opposite side c.

 

 

Solving for a Side: Side – Angle – Side

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Since we’re solving for a side, put the unknown side on the top-left fraction, then solve for b.

    \[\frac{b}{\sin(110)}=\frac{10}{\sin(45)}\]

    \[b=\frac{10\sin(110)}{\sin(45)}\approx 13.3\]

 

Solving for an Angle: Side – Side – Angle

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Since we’re solving for an angle we must use an inverse sine to finish the problem. Just like above, we put the unknown in the top left of the fraction.

    \[\frac{\sin(C)}{9}=\frac{\sin(62.2)}{8}\]

    \[\sin(C)=\frac{9\sin(62.2)}{8}\]

    \[C=\sin^{-1}\left(\frac{9\sin(62.2)}{8}\right)\approx84.36\]

 

Proof

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\sin(A)=\frac{d}{b}                          \sin(B)=\frac{d}{a} \\

d=b\sin(A)                                     d=a\sin(B) \\

b\sin(A)=a\sin(B)

\frac{b}{\sin(B)}=\frac{a}{\sin(A)}