在△ABC中,角A,B,C为三个内角,已知A＝45°,cosB＝4/5.①求cosC的值.▁▂▃▄▅▆▇█在△ABC中,角A,B,C为三个内角,已知A＝45°,cosB＝4/5.①求cosC的值.②若BC＝10.D为AB的中点,求CD的长

Question

在△ABC中,角A,B,C为三个内角,已知A＝45°,cosB＝4/5.①求cos C的值.
▁▂▃▄▅▆▇█在△ABC中,角A,B,C为三个内角,已知A＝45°,cosB＝4/5.①求cos C的值.
②若BC＝10.D为AB的中点,求CD的长

Answer 1

cosB=4/5
sinB=3/5
sinA=sin45=√2/2
cosA=cos45=√2/2
cosC=cos(180-A-B)=-cos(A+B)
=-cosAcosB+sinAsinB
=-(√2/2)(4/5)+(√2/2)(3/5)
=-√2/10
AC/sinB=BC/sinA
AC=BC*sinB/sinA=10(3/5)/(√2/2)=6√2
AB/sinC=BC/sinA
AB=BC*sinC/sinA=10*(7√2/10)/(√2/2)=14
AD=AB/2=14/2=7
CD^2=AC^2+AD^2-2AC*AD*cosA
=72+49-2*(6√2)*7(√2/2)
=37
CD=√37