EVALUATE THE TRIGONOMETRIC FUNCTIONS USING COMPOUND ANGLE FORMULA

Compound Angle Formulas

sin (A + B) = sin A cos B + cos A sin B

sin (A - B) = sin A cos B - cos A sin B

cos (A + B) = cos A cos B - sin A sin B

cos (A - B) = cos A cos B + sin A sin B

If sin x = 4/5 and sin y = -12/13, 0 < x < π/2, 3π/2 < y < 2π, evaluate

Problem 1 :

cos (x + y)

Solution :

Given,  sin x = 4/5 and sin y = -12/13

To find cos (x + y) :

cos x = 1 - sin2x= 1 - 452= 1 - 1625= 25 - 1625= 925cos x = 35
cos y = 1 - sin2 y= 1 - -12132= 1 - 144169= 169 - 144169= 25169cos y = 513

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

cos (x + y)=35 ⋅ 513 - 45 ⋅ -1213 = 313 + 4865= 1565 + 4865= 6365cos (x + y) = 6365

Problem 2 :

sin (x + y)

Solution :

Given,  sin x = 4/5 and sin y = -12/13

To find sin (x + y) :

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

cos x = 1 - sin2x= 1 - 452= 1 - 1625= 25 - 1625= 925cos x = 35
cos y = 1 - sin2 y= 1 - -12132= 1 - 144169= 169 - 144169= 25169cos y = 513
sin (x + y)=45 ⋅ 513 + 35 ⋅ -1213 = 413 - 3665= 2065 - 3665= -1665sin (x + y) = -1665

Problem 3 :

cos (x - y)

Solution :

Given,  sin x = 4/5 and sin y = -12/13

To find cos (x - y) :

cos x = 1 - sin2x= 1 - 452= 1 - 1625= 25 - 1625= 925cos x = 35
cos y = 1 - sin2 y= 1 - -12132= 1 - 144169= 169 - 144169= 25169cos y = 513

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

cos (x - y)=35 ⋅ 513 + 45 ⋅ -1213 = 313 - 4865= 1565 - 4865= -3365cos (x - y) = -3365

Problem 4 :

sin (x - y)

Solution :

Given,  sin x = 4/5 and sin y = -12/13

To find sin (x - y) :

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

cos x = 1 - sin2x= 1 - 452= 1 - 1625= 25 - 1625= 925cos x = 35
cos y = 1 - sin2 y= 1 - -12132= 1 - 144169= 169 - 144169= 25169cos y = 513
sin (x - y)=45 ⋅ 513 - 35 ⋅ -1213 = 413 + 3665= 2065 + 3665= 5665sin (x - y) = 5665

Problem 5 :

tan (x + y)

Solution :

Given,  sin x = 4/5 and sin y = -12/13

To find tan (x + y) :

tan (x + y) = tan x + tan y1 - tan x tan y
tan x = sin xcos x
cos x = 1 - sin2x= 1 - 452= 1 - 1625= 25 - 1625= 925cos x = 35
tan x = 4535= 45 × 53tan x = 43
 tan y = sin ycos y
cos y = 1 - sin2 y= 1 - -12132= 1 - 144169= 169 - 144169= 25169cos y = 513
tan y = -1213513= -1213 × 135tan y = -125
tan (x + y) = 43 - 1251 - 43 -125= 20 - 36151 + 4815 = -161515 + 4815 = -16156315 = -1615 × 1563tan (x + y)= -1663

Problem 6 :

tan (x - y)

Solution :

Given,  sin x = 4/5 and sin y = -12/13

To find tan (x - y) :

tan (x - y) = tan x - tan y1 + tan x tan y
tan x = sin xcos x
cos x = 1 - sin2x= 1 - 452= 1 - 1625= 25 - 1625= 925cos x = 35
tan x = 4535= 45 × 53tan x = 43
 tan y = sin ycos y
cos y = 1 - sin2 y= 1 - -12132= 1 - 144169= 169 - 144169= 25169cos y = 513
tan y = -1213513= -1213 × 135tan y = -125
tan (x - y) = 43 + 1251 + 43 -125= 20 + 36151 - 4815 = 561515 - 4815 = 5615-3315 = 5615 × -1533tan (x - y)= -5633

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