To Prove:
a cosA+b cosB+c cosC=2 sin B sin C

Asked by prasantaratha.lic | 24th Nov, 2016, 10:46: AM

Expert Answer:

begin mathsize 16px style The space question space should space be space to space prove space that colon
straight a space cosA plus straight b space cosB plus straight c space cosC equals 2 space straight a space sinB space sinC

Using space the space Sine space rule space we space get comma
straight a over sinA equals straight b over sinB equals straight c over sinC equals straight k space left parenthesis say right parenthesis
rightwards double arrow straight a equals straight k space sinA comma space straight b equals straight k space sinB comma space straight c equals straight k space sinC
Consider comma space LHS
equals straight a space cosA plus straight b space cosB plus straight c space cosC
equals straight k space sinA space cosA plus straight k space sinB space cosB plus straight k space sinC space cosC
equals 1 half straight k space open square brackets 2 sinA space cosA plus 2 space sinB space cosB plus space 2 sinC space cosC close square brackets
equals 1 half straight k space open square brackets sin 2 straight A plus space sin 2 straight B plus space sin 2 straight C close square brackets
equals 1 half straight k space open square brackets 2 sin left parenthesis straight A plus straight B right parenthesis space cos left parenthesis straight A minus straight B right parenthesis space plus space 2 sinC space cosC close square brackets
equals straight k open square brackets sin left parenthesis straight A plus straight B right parenthesis space cos left parenthesis straight A minus straight B right parenthesis space plus space sinC space cosC close square brackets
equals straight k open square brackets sin left parenthesis straight pi minus straight C right parenthesis space cos left parenthesis straight A minus straight B right parenthesis space plus space sinC space cosC close square brackets
equals straight k open square brackets sinC space cos left parenthesis straight A minus straight B right parenthesis space plus space sinC space cosC close square brackets
equals straight k open square brackets sinC space open curly brackets cos left parenthesis straight A minus straight B right parenthesis space plus space cosC close curly brackets close square brackets
equals straight k open square brackets sinC space open curly brackets cos left parenthesis straight A minus straight B right parenthesis space plus space cos open parentheses straight pi minus left parenthesis straight A plus straight B right parenthesis close parentheses close curly brackets close square brackets
equals straight k open square brackets sinC space open curly brackets cos left parenthesis straight A minus straight B right parenthesis space plus space cos left parenthesis straight A plus straight B right parenthesis close curly brackets close square brackets
equals straight k open square brackets sinC space left parenthesis 2 sinAsinB right parenthesis close square brackets
equals 2 straight k space sinA space sinB space sinC
equals 2 open parentheses straight k space sinA close parentheses space sinB space sinC
equals 2 space straight a space sinB space sinC
Hence space proved. end style

Answered by Rebecca Fernandes | 24th Nov, 2016, 02:46: PM