082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

3.6 Trigonometric identities 129
Example3.7 Show that
( ) ( ) A +B A −B
sinA+sinB=2sin cos
2 2
Solution Consider the identities
sin(C +D) = sinCcosD +sinDcosC
sin(C −D) = sinCcosD −sinDcosC
Byadding these identities weobtain
sin(C +D) +sin(C −D) =2sinCcosD
We now make the substitutionsC +D =A,C −D =Bfromwhich
C = A +B , D = A −B
2 2
Hence
( ) ( ) A +B A −B
sinA+sinB=2sin cos
2 2
TheresultofExample3.7isoneofmanysimilarresults.ThesearelistedinTable3.2.
Table3.2
Further trigonometricidentities
( A +B
sinA+sinB =2sin
2
( A −B
sinA−sinB =2sin
2
( A +B
cosA+cosB=2cos
2
( A +B
cosA−cosB=−2sin
2
)
cos
)
cos
)
cos
( ) A −B
2
( ) A +B
2
( ) A −B
2
( ) A −B
)
sin
2
Example3.8 Simplify
sin70 ◦ −sin30 ◦
cos50 ◦
Solution We note that the numerator, sin70 ◦ − sin30 ◦ , has the form sinA − sinB. Using the
identityfor sinA −sinBwithA = 70 ◦ andB = 30 ◦ wesee
( ) ( )
70
sin70 ◦ −sin30 ◦ ◦ −30 ◦ 70 ◦ +30 ◦
= 2sin cos
2 2
= 2sin20 ◦ cos50 ◦