FIND THE VALUE OF SIN X PLUS Y WHEN SIN X AND COS Y ARE GIVEN

Compound Angle Formula in Trigonometry

Formula for sin (x+y) and sin (x-y)

sin (x + y)  =  sin x cos y + cos x sin y

sin (x - y)  =  sin x cos y - cos x sin y

Formula for cos (x+y) and cos (x-y)

cos (x + y)  =  cos x cos y - sin x sin y

cos (x - y)  =  cos x cos y + sin x sin y

Formula for tan (x+y) and tan (x-y)

tan (x + y)  =  (tan x + tan y) / (1 - tan x tan y)

tan (x - y)  =  (tan x - tan y) / (1 + tan x tan y)

Problem :

If sin x = 15/17 and cos y = 12/13 , 0 < x < π/2 , 0 < y < π/2, find the values of

(i) sin(x + y)         (ii) cos(x − y)          (iii) tan(x + y).

Solution :

(i) sin (x + y) 

Formula for sin (x + y) is sin x cos y + cos x sin y.Now we have to find the values of cos x and sin y.

sin x = 15/17 

cos x  =  √(1 - sin2x)  

  =  √(1 - (15/17)2

  =  √(1 - (225/289))

  =  √(289 - 225)/289 

  =  √(64/289) 

cos x  =  8/17

cos y = 12/13 

sin y  =  √(1 - cos2y)

  =  √(1 - (12/13)2

  =  √(1 - (144/169))

  =  √(169 - 144)/169 

  =  √(25/169) 

sin y  =  5/13

sin(x + y)  =  sin x cos y + cos x sin y

  =  (15/17)  (12/13) + (8/17) ⋅ (5/13)

  =  (180/221) + (40/221)

  =  (180 + 40)/221

  =  220/221

Hence the value of sin (x + y) is 220/221.

(ii) cos(x − y) 

Formula for cos (x - y) is cos x cos y + sin x sin y. 

cos (x - y)   =  (8/17)  (12/13) + (15/17) ⋅ (5/13)

  =  (96/221) + (75/221)

  =  (96 + 75)/221

=  171/221

Hence the value of cos (x - y) is 171/221.

(iii) tan (x + y)

Formula for tan (x + y)  =  (tan x + tan y)/(1 - tan x tan y)

Now we have to find the values of tan x and tan y.

tan x  =  sin x / cos x

  =  (15/17) / (8/17)

  =  (15/17)  (17/8)

tan x  =  15/8

tan y  =  sin y / cos y

  =  (5/13) / (12/13)

  =  (5/13)  (13/12)

tan y  =  5/12

tan (x + y)  =  (15/8) + (5/12)/(1 - (15/8)(5/12))

  =  [(180 + 40)/96] / [1 - (75/96)]

  =  (220/96) / (21/96)

  =   220/21

Hence the value of tan (x + y) is 220/21.

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