# Suppose that m /_BCA=70^@, and x=33 cm and y = 47 cm. What is the degree measure of /_ABC? 01510101651.png

Non-right triangles and trigonometry
Suppose that $$\displaystyle{m}\angle{B}{C}{A}={70}^{\circ}$$, and x=33 cm and y = 47 cm. What is the degree measure of $$\displaystyle\angle{A}{B}{C}$$?

2020-12-28
$$\displaystyle{A}{B}=\sqrt{{{A}{C}^{{2}}+{B}{C}^{{2}}-{\left({2}\cdot{A}{C}\cdot{B}{C}{\cos}\angle{B}{C}{A}\right)}}}$$
$$\displaystyle{A}{B}=\sqrt{{{47}^{{2}}+{33}^{{2}}-{\left({2}\cdot{47}\cdot{33}{\cos{{70}}}^{\circ}\right)}}}$$
$$\displaystyle{A}{B}=\sqrt{{{2237.05}}}$$
$$\displaystyle{A}{B}={47.29}$$
Using the cosine law,
$$\displaystyle\angle{A}{B}{C}={{\cos}^{{-{{1}}}}{\left(\frac{{{B}{C}^{{2}}+{A}{B}^{{2}}-{A}{C}^{{2}}}}{{{2}\cdot{B}{C}\cdot{A}{B}}}\right)}}$$
$$\displaystyle\angle{A}{B}{C}={{\cos}^{{-{{1}}}}{\left(\frac{{{33}^{{2}}+{47.29}^{{2}}-{47}^{{2}}}}{{{2}\cdot{33}\cdot{47.29}}}\right)}}$$
$$\displaystyle\angle{A}{B}{C}={{\cos}^{{-{{1}}}}{\left({0.3576}\right)}}$$
$$\displaystyle\angle{A}{B}{C}={{\cos}^{{-{{1}}}}=}{69.04}^{\circ}$$