Answer:
[tex]these \: angles \: are \: \boxed{ \to} \\ \boxed{X = \:60.08 } \\ \\ \boxed{Y = \:72.45 }\\ \\ \boxed{Z =47.47}[/tex]
Step-by-step explanation:
[tex]....................................................... \\ well, \: there \: are \: no \: \: angles \: at \: all \to \\ but \: am \: going \: to \: sho w\: you \\how \: all \: these \: angles \: can \: be \: gotten. \\ ........................................................ \\ but \: to \: do\: this : \: we \: will \: apply \: the \\ cosine \: rule \: in \: the \: general \: form : \\ \boxed{\cos(C) = \frac{{a}^{2} + {b}^{2} - {c}^{2}}{2ab} } \\ where \to \\ \boxed{C} = the \: missing \: angle \\ \boxed{c} = the \: side \: opposite \: to \: angle \: \boxed{C} \\ note : \to \: that \: \boxed{C} \: can \: always \: be \: applied \: to \: any \: \\ missing \: angle.\\ \\ \boxed{ lets \: first\: find \: angle \: X}: \to \\ for \: angle \boxed{X} \: the \: cosine \: rule \: can \: be \: written \: as : \\ \boxed{\cos(X) = \frac{{y}^{2} + {z}^{2} - {x}^{2}}{2yz} } \\ where : x \: y \: and \: z \: are \: the \: given \: sides \to \\ x = 20 \\ y = 22 \\ z = 17\\then \: \to \\ \cos(X) = \frac{{22}^{2} + {17}^{2} - {20}^{2}}{2(22)(17)} \\ \\ \cos(X) = \frac{373}{748}\\ \cos(X) = 0.4986631016 \\X = \ cos {}^{ - 1} (0.4986631016) \\ \boxed{X = \:60.08 } \\ .......................................................... \\ \\ \boxed{ lets \: find \: angle \: Y \: next}: \to\\for \: angle \boxed{Y} \: the \: cosine \: rule \: can \: be \: written \: as : \\ \boxed{\cos(Y) = \frac{{x}^{2} + {z}^{2} - {y}^{2}}{2xz} } \\ where : x \: y \: and \: z \: are \: the \: given \: sides \to \\ x = 20 \\ y = 22 \\ z = 17\\then \: \to \\ \cos(Y) = \frac{{20}^{2} + {17}^{2} - {22}^{2}}{2(20)(17)} \\ \\ \cos(Y) = \frac{205}{680}\\ \cos(Y) = 0.3014705882 \\Y = \ cos {}^{ - 1} (0.3014705882) \\ \boxed{Y = \:72.45 } \\ .......................................................... \\ \\ \boxed{ lets \: finaly \: find \: angle \: Z}: \to \\ to \: do \: this \: you \: can \: aswell \: use \: the \: cosine \: rule : \\ but \: lets \: try \: some \: new \: laws \to \\ X + Y + Z = 180 \: \to \: (sum \:o f \: angles) \\ 60.08 + 72.45 + Z = 180 \\ Z = 180 - (60.08 + 72.45 ) \\ Z = 180 -132.53 \\ \boxed{Z =47.47}[/tex]
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