Bài 4 trang 95 Tài liệu dạy – học Toán 9 tập 1

Đề bài

Giải tam giác ABC vuông tại A, biết rằng :

a) \(b = 8cm,\widehat C = {60^o}\);

b) \(c = 12cm,\widehat C = {30^o}\);

c) \(a = 12cm,\widehat C = {45^o}\);

d) \(a = 10cm;\widehat B = {55^o}\);

e) \(c = 42cm,b = 36cm\).

Lời giải chi tiết

a) \(b = 8cm,\widehat C = {60^o}\);

\(\begin{array}{l}c = b.\tan \widehat C = 8.\tan {60^o} \approx 13,86\,\,(cm)\\b = a.\cos \widehat C \Rightarrow a = \dfrac{b}{{\cos \widehat C}} = \dfrac{8}{{\cos {{60}^o}}} = 16\,\,(cm)\\\widehat B = {90^o} - \widehat C = {90^o} - {60^o} = {30^o}\end{array}\)

b) \(c = 12cm,\widehat C = {30^o}\);

\(\begin{array}{l}b = c.\cot \widehat C = 12.\cot {30^o} \approx 20,78\,\,(cm)\\c = a.\sin \widehat C \Rightarrow a = \dfrac{c}{{\sin \widehat C}} = \dfrac{{12}}{{\sin {{30}^o}}} = 24\,\,(cm)\\\widehat  B = {90^o} - \widehat  C = {90^o} - {30^o} = {60^o}\end{array}\)

c) \(a = 12cm,\widehat C = {45^o}\);

\(\begin{array}{l}c = a.\sin \widehat C = 12.\sin {45^o} \approx 8,49\,\,(cm)\\b = a.\cos \widehat C = 12.\cos {45^o} \approx 8,49\,\,(cm)\\\widehat B = {90^o} - \widehat C = {90^o} - {45^o} = {45^o}\end{array}\)

d) \(a = 10cm;\widehat B = {55^o}\);

\(\begin{array}{l}b = a.\sin \widehat B = 10.\sin {55^o} \approx 8,19\,\,(cm)\\c = a.\cos \widehat B = 10.\cos {55^o} \approx 5,74\,\,(cm)\\\widehat C = {90^o} - \widehat B = {90^o} - {55^o} = {35^o}\end{array}\)

e) \(c = 42cm,b = 36cm\).

Áp dụng định lý Pythagore: \(a = \sqrt {{b^2} + {c^2}}  = \sqrt {{{42}^2} + {{36}^2}}  \approx 55,32\,\,(cm)\)

\(\begin{array}{l}\tan \widehat B = \dfrac{b}{c} = \dfrac{6}{7} \Rightarrow \widehat B \approx {40^o}36'\\\widehat C = {90^o} - \widehat B \approx {49^o}24'\end{array}\)