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Trigonometric Functions of Sum and Difference of Two Angles (Part I)

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Trigonometric Functions of Sum and Difference of Two Angles (Part I) - Lesson Summary

Consider a unit circle such that its centre is at the origin. Let A, B, C and D be points on the circle such that angle AOB is x, angle BOC is y and angle AOD is -y.

P(a, b) = P(cos x, sin x)

cos x  = Adjacent side to x Hypotenuse    =  a 1    =  a
sin x  = Opposite side to x Hypotenuse    =  b 1    =  b

cos(- x) = cos x and sin(- x) = - sin x

A = (1, 0) [âˆµ Radius of the circle is 1 unit]

B = (cos x, sin x)

C = [cos(x + y), sin (x + y)]

D = [cos(-y , sin (-y))] = (cos y, -sin y)

In Î”BOD and Î”AOC,

OB = OC [âˆµ Radii of the circle are equal]

OD = OA [âˆµ Radii of the circle are equal]

Also, âˆ BOD = y + âˆ COD = âˆ AOC

Î”BOD â‰… Î”AOC                  (By SAS congruence)

â‡’ BD = AC

â‡’ BD 2 = AC 2

â‡’ (cos x - cos y) 2 + (sin x + sin y) 2 = [cos(x + y) â€“ 1] 2 + [sin(x + y) â€“ 0] 2    (âˆµ Distance between two points is ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2  )
â‡’ cos 2x + cos 2y - 2 cos x cos y + sin 2x + sin 2y + 2 sin x sin y

â‡’ cos 2(x + y) - 2 cos (x + y) + 1 + sin 2(x + y)

â‡’ -2 cos(x + y) = -2(cos x cos y â€“ sin x sin y)     (Since cos 2 Î¸ + sin 2 Î¸ = 1)

â‡’ cos(x + y) = cos x cos y â€“ sin x sin y

cos(x + y) = cos x cos y â€“ sin x sin y

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

Replacing y by â€“ y, cos(x - (â€“y)) = cos x cos (â€“y) + sin x sin (â€“y)

Since cos(â€“ x) = cos x and sin(â€“ x) = â€“ sin x

cos(x - y) = cos x cos y + sin x sin y  ______ (2)

cos ( Ï€ 2   - y) = sin x

cos ( Ï€ 2   - x) = cos  Ï€ 2 cos x + sin Ï€ 2 sin x

= 0 × cos x + 1 × sin x

cos ( Ï€ 2   - x) = sin x

sin ( Ï€ 2   - x) = cos x

sin ( Ï€ 2   - x) = cos ( Ï€ 2   â€“ ( Ï€ 2   â€“ x)) = cos x

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

sin(x + y) = cos ( Ï€ 2   â€“ ( x + y ))

= cos (( Ï€ 2   â€“ x ) â€“ y ))

=  cos( Ï€ 2   â€“ x ) cos y + sin( Ï€ 2   â€“ x ) sin y

â‡’ sin (x + y) = sin x cos y + cos x sin y

sin (x â€“ y) = sin x cos y â€“ cos x sin y

Replacing y by â€“ y in the equation

sin(x + (-y)) = sin x cos (â€“ y) + cos x sin (â€“ y)

since cos (â€“ x) = cos x and sin (â€“ x) = â€“ sin x

sin(x â€“ y) = sin x cos y â€“ cos x sin y

cos( Ï€ 2   + x ) = â€“ sin x

We know that  cos( Ï€ 2   â€“ x ) = sin x

Replacing x by â€“ x, we get

cos( Ï€ 2   â€“ (â€“ x) ) = sin (â€“ x)

â‡’ cos( Ï€ 2   + x ) = â€“ sin x

sin( Ï€ 2   + x ) = cos x

We know that sin( Ï€ 2   â€“ x ) = cos x

Replacing x by â€“ x, we have sin( Ï€ 2   â€“ (â€“ x) ) = cos (â€“ x)

â‡’ sin( Ï€ 2   + x ) = cos x

Similarly, we can prove that

cos (Ï€ â€“ x) = â€“ cos x
sin (Ï€ â€“ x) =  sin x
cos (Ï€ + x) = â€“ cos x
sin (Ï€ + x) = â€“ sin x
cos (2Ï€ â€“ x) = cos x
cos (2Ï€ â€“ x) = â€“ sin x

Similar results for tan x, cot x, sec x and cosec x can be obtained from the results of sin x and cos x.  These results are shown in the following table.
Ï€ 2 - x   Ï€ 2 + x   Ï€  - x Ï€  + x   3Ï€ 2 - x   3Ï€ 2 + x   2Ï€  - x   2Ï€  + x  sin  cos x  cos x  sin x  - sin x  - cos x  - cos x  - sin x  sin x  cos  sin x  - sin x  - cos x  - cos x  - sin x  sin x  cos x  cos x  tan  cot x  - cot x  - tan x  tan x  cot x  - cot x  - tan x  tan x  cot  tan x  - tan x  - cot x  cot x  tan x  - tan x  - cot x  cot x  sec  cosec x  - cosec x  - sec x  - sec x  - cosec x  sec x  sec x  sec x  cosec  sec x  sec x  cosec x  - cosec x  - sec x  - sec x  - cosec x cosec x
Tan and Cot functions of the sum and differences of two angles:

Cos Î¸ = 0 implies that Î¸ = (2n+1)  Ï€ 2  , where n is any integer.

tan Î¸ =  sin Î¸ cos Î¸ is defined if Î¸ â‰  (2n+1) Ï€/2

tan(x + y) = tan x + tan y 1 - tan x tan y  if x, y, x + y â‰  (2n+1) Ï€/2

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

= sin x cos y + cos x sin y cos x cosy - sin x siny

= sin x cos y cos x cos y + cos x sin y sin x sin y cos x cos y cos x cos y - sin x sin y cos x cos 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

Replacing y by â€“ y, we have, tan(x + (-y)) = tan x + tan (-y) 1 - tan x tan (-y)

â‡’ tan(x - y) = tan x - tan y 1 + tan x tan y

Sin Î¸ = 0 implies that Î¸ =n Ï€, where n is any integer.

cot Î¸ = cos Î¸ sin Î¸  is defined if Î¸ â‰  n Ï€

cot (x + y) = cot x cot y - 1 cot y + cot x

if x, y, x + y â‰  n Ï€

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

Dividing both numerator and denominator by sin x sin y,

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

â‡’  cot (x + y) = cot x cot y - 1 cot y + cot x

cot (x - y) = cot x cot y + 1 cot y - cot x

Replacing y by â€“ y, cot (x - (-y)) = cot x cot ( -y) - 1 cot (-y) + cot x

cot (x - y) = - cot x cot y - 1 - cot y + cot x

â‡’ cot (x - y) = cot x cot y + 1 cot y - cot x