Áp dụng định lý cosin trong `ΔABC` ta có:

`BC^2=AB^2 +AC^2 -2.AB.AC.cosA`

`BC^2=6^2 +7^2-2.6.7.cos30^@`

`BC^2=36+49-84. sqrt(3)/2`

`BC^2=85-42sqrt(3)`

`=> BC=sqrt(85-42sqrt(3))`

Áp dụng định lí `sin` trong `ΔABC` ta có:

`(BC)/sin hat(A)=(AB)/sin hat(C)`

`->sin hat(C)=AB.sin hat(A)/(BC)`

`->sin hat(C)=6.sin(30^@)/sqrt(85-42sqrt(3))`

`->sin hat(C)=6.(1/2)/sqrt(85-42sqrt(3))`

`->sin hat(C)=3/sqrt(85-42sqrt(3))`

`=>hat(C)~~59^@`

Áp dụng định lí `sin` cho `ΔABC` ta có:

`(BC)/sin hat(A)=(AC)/sin hat(B)`

`=>sin hat(B)=AC.sin hat(A)/(BC)`

`=>sin hat(B)=7.sin(30^@)/sqrt(85-42sqrt(3))`

`=>sin hat(B)=7.(1/2)/sqrt(85-42sqrt(3))`

`=>sin hat(B)=7/(2sqrt(85-42sqrt(3)))`

`=>hat(B)~~91^@`

Vậy `BC=sqrt(85-42sqrt(3)),hat(B)~~91^@,hat(C)~~59^@`

________

`***` Định lí cosin trong `ΔABC`:

`a^2=b^2+c^2-2.b.c.cos hat(A)`

`b^2=a^2+c^2-2.a.c.cos hat(B)`

`c^2=a^2+b^2-2.a.b.cos hat(C)`

`***` Định lí sin trong `ΔABC`:

`a/sin hat(A)=b/sin hat(B)=c/sin hat(C)=2.R`

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