sin α + sin β = 3 and cos α + cos β = 4
(i) The value of sin ( α + β ) is :
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(i)
sin α + sin β = 3 ..(i) & cos α + cos β = 4 ...(ii)
(sin α + sin β) 2 + ( cos α + cos β) 2 = 16 + 9
⇒ 1+1+2cos( α – β )= 25 ⇒ cos( α – β )=23/2 ...(iii)
Also squaring & subtract (i) form (ii)
cos2 α + cos2 β + 2cos( α + β ) = 7
2cos( α + β ) cos( α – β ) + 2 cos( α + β ) = 7..(iv)
put (iii) in (iv) 25 cos( α + β ) = 7 ⇒ cos ( α + β ) = 7/25
∴ sin( α + β ) = 24/25 & tan ( α+β ) = 24/7
(ii)
∴ sin α + sin β = 3 ..(i) & cos α + cos β = 4 ...(ii)
(sin α + sin β) 2 + ( cos α + cos β) 2 = 16 + 9
⇒ 1+1+2cos( α – β )= 25 ⇒ cos( α – β )=23/2 ...(iii)
Also squaring & subtract (i) form (ii)
cos2 α + cos2 β + 2cos( α + β ) = 7
2cos( α + β ) cos( α – β ) + 2 cos( α + β ) = 7..(iv)
put (iii) in (iv) 25 cos( α + β ) = 7 ⇒ cos ( α + β ) = 7/25
∴ sin( α + β ) = 24/25 & tan ( α+β ) = 24/7
(iii)
∴ sin α + sin β = 3 ..(i) & cos α + cos β = 4 ...(ii)
(sin α + sin β) 2 + ( cos α + cos β) 2 = 16 + 9
⇒ 1+1+2cos( α – β )= 25 ⇒ cos( α – β )=23/2 ...(iii)
Also squaring & subtract (i) form (ii)
cos2 α + cos2 β + 2cos( α + β ) = 7
2cos( α + β ) cos( α – β ) + 2 cos( α + β ) = 7..(iv)
put (iii) in (iv) 25 cos( α + β ) = 7 ⇒ cos ( α + β ) = 7/25
∴ sin( α + β ) = 24/25 & tan ( α+β ) = 24/7
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