Notes: `A = 120^@, a = 25, b = 24` Use the law of sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answer...

Wednesday, 18 November 2015

$\displaystyle{A}={{120}}^{\circ},{a}={25},{b}={24}$ Use the law of sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answer...

Given: $\displaystyle{A}={{120}}^{\circ},{a}={25},{b}={24}$

The Law of Sines: $\displaystyle\frac{{a}}{{\sin{{\left({A}\right)}}}}=\frac{{b}}{{\sin{{\left({B}\right)}}}}=\frac{{c}}{{\sin{{\left({C}\right)}}}}$

$\displaystyle\frac{{25}}{{\sin{{\left({120}\right)}}}}=\frac{{24}}{{\sin{{\left({B}\right)}}}}=\frac{{c}}{{\sin{{\left({C}\right)}}}}$

$\displaystyle{\sin{{\left({B}\right)}}}=\frac{{{24}{\sin{{\left({120}\right)}}}}}{{25}}={.8313}$

$\displaystyle{B}={\arcsin{{\left({.8313}\right)}}}$

$\displaystyle{B}={56.24}$

$\displaystyle{C}={180}-{120}-{56.24}$

$\displaystyle{C}={3.76}$

$\displaystyle\frac{{25}}{{\sin{{\left({120}\right)}}}}=\frac{{c}}{{\sin{{\left({3.76}\right)}}}}$

$\displaystyle{c}=\frac{{{25}{\sin{{\left({3.76}\right)}}}}}{{\sin{{\left({120}\right)}}}}$

$\displaystyle{c}={1.89}$

Given: $\displaystyle{A}={{120}}^{\circ},{a}={25},{b}={24}$

The Law of Sines: $\displaystyle\frac{{a}}{{\sin{{\left({A}\right)}}}}=\frac{{b}}{{\sin{{\left({B}\right)}}}}=\frac{{c}}{{\sin{{\left({C}\right)}}}}$

$\displaystyle\frac{{25}}{{\sin{{\left({120}\right)}}}}=\frac{{24}}{{\sin{{\left({B}\right)}}}}=\frac{{c}}{{\sin{{\left({C}\right)}}}}$

$\displaystyle{\sin{{\left({B}\right)}}}=\frac{{{24}{\sin{{\left({120}\right)}}}}}{{25}}={.8313}$

$\displaystyle{B}={\arcsin{{\left({.8313}\right)}}}$

$\displaystyle{B}={56.24}$

$\displaystyle{C}={180}-{120}-{56.24}$

$\displaystyle{C}={3.76}$

$\displaystyle\frac{{25}}{{\sin{{\left({120}\right)}}}}=\frac{{c}}{{\sin{{\left({3.76}\right)}}}}$

$\displaystyle{c}=\frac{{{25}{\sin{{\left({3.76}\right)}}}}}{{\sin{{\left({120}\right)}}}}$

$\displaystyle{c}={1.89}$

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