6.5 Formulae of sin (A ± B), cos (A ± B), tan (A ± B), sin 2A, cos 2A, tan 2A
(A) Compound Angles Formulae:
(B) Double Angle Formulae:
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(C) Half Angle Formulae:
• sinA=2sinA2cosA2 • cosA=sin2A2−cos2A2 cosA=2cos2A2−1 cosA=1−2cos2A2 • tanA=2tanA21−tan2A2
5.5.1 Proving Trigonometric Identities using Addition Formula and Double Angle Formulae
Example 1:
(a) sin(A+B)−sin(A−B)cosAcosB=2tanB(b) cos(A+B)sinAcosB=cotA−tanB(c) tan(A+45o)=sinA+cosAcosA−sinA
Solution:
(a)
LHS=sin(A+B)−sin(A−B)cosAcosB=(sinAcosB+cosAsinB)−(sinAcosB−cosAsinB)cosAcosB=2cosAsinBcosAcosB=2sinBcosB=2tanB=RHS (proven)
(b)
LHS=cos(A+B)sinAcosB=cosAcosB−sinAsinBsinAcosB=cosAcosBsinAcosB−sinAsinBsinAcosB=cosAsinA−sinBcosB=cotA−tanB=RHS(proven)
(c)
LHS=tan(A+45o)=tanA+tan45o1−tanAtan45o=tanA+11−tanA←tan45o=1=sinAcosA+11−sinAcosA=sinA+cosAcosA×cosAcosA−sinA=sinA+cosAcosA−sinA=RHS(proven)
(b)
LHS=cos(A+B)sinAcosB=cosAcosB−sinAsinBsinAcosB=cosAcosBsinAcosB−sinAsinBsinAcosB=cosAsinA−sinBcosB=cotA−tanB=RHS(proven)
(c)
LHS=tan(A+45o)=tanA+tan45o1−tanAtan45o=tanA+11−tanA←tan45o=1=sinAcosA+11−sinAcosA=sinA+cosAcosA×cosAcosA−sinA=sinA+cosAcosA−sinA=RHS(proven)