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