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4.2 The Addition Formulas (Identities)

In this part, we will study the trigonometric ratios (functions) of sum and difference of two angles.

Verification

1) For all real values of a and b cos (a - b) = cos a cos b + sin a sin b.

Consider a unit circle as shown in the figure. Let R, P, Q, be three points on it such that Ð ROP = a, Ð ROQ = b so that ÐQOP = a - b. Set the centre of the circle at the origin of the rectangular Cartesian system. Let x- axis be along OR. Draw Y- axis.

Here we have OR = OP = OQ = 1

\ R º (1,0), P º ( cos a , sin a ) and Q º (cos b, sin b)    ..... (Using x = r cos q, y = r sin q i.e. changing to polar coordinates)

\ By distance formula,

PQ2 = (cos a - cos b)2 + (sin a - sin b)2
= (cos2 a + sin 2 a ) + (cos2 b + sin2 b) - 2 (cos a cos b + sin a sin b)

= 1 + 1 - 2(cos a cos b + sin a sin b )

\ PQ2= 2 -2 (cos a cos b + sin a sin b)     .......(1)

Take point S on the circle so that ÐROS = a-b
\ S º [cos (a - b), sin (a - b)]

Then by distance formula,
RS2 = [cos (a- b) -1 ]2 + [sin (a -b )]2
= cos2 (a - b) + sin2(a - b) -2 cos (a - b)
RS2 = 2 - 2 cos (a - b )                      ........(2)

Since ÐQOP = ÐROS, their corresponding arcs PQ and RS are also equal which implies that the corresponding chords PQ and RS are also equal.

\ from (1) and (2) we get

2 - 2 cos (a - b) = 2 - 2 (cos a cos b + sin a sin b )

put a = x and b = -y in
cos (x - y ) = cos x cos y + sin x sin y, x, y Î R
We get cos (a + b) = cos a cos (-b) + sin a sin ( -b)
but cos (-b) = cos b and sin (-b) = - sin b

Reduction to Functions of Positive Acute Angles And Negative Acute Angles.

Note that * 1) sin (n 900 ± a) = ± cos a, n Î N and n is odd
* 2) cos (n 900 ± a) = ± sin a, n Î N and n is odd
* 3) sin (n 900 ± a) = ± sin a, n Î N and n is even
* 4) cos (n 900 ± a) = ± cos a, n Î N and n is even

The sign on R.H.S is determined by the Quad. of the angle in L.H.S using ` A S T C'

Example 1 Evaluate sin 800cos 1300+ cos 800 sin 1300

Solution Let a = 800and b = 1300 then sin 800 cos 1300 + cos 800sin 1300 = sin a cos b + cos a sin b
= sin (a + b)
= sin (800+1300)
= sin (2100)
= sin (1800 + 300)
= - sin 300

Example 2 Find the value of i) cos 4800 ii) tan (- 9450) iii) sin 46200

Solution

  1. cos 4800= cos ( 5 90 0 + 300) = - sin 300 = -1/2
    As angle in Quad II, cosine ratio is negative.     .... [Note : 4800 = 3600 + 1200 coterminal to 1200. Hence in Quad. II ]

  2. tan ( -9450) = - tan (9450)    .... \ tan (- q) = - tan q
    = - tan (10 90 0 + 450)
    = - tan 450    .... \ tan (n900+ a ) = + tan a for n is even and angle is in Quadrant III.
    = -1

[Note : 945 = 2 360 + 225 coterminal to 225. Hence in Quad. III]

Example 3

Express of the following in terms of a function of :
(i) sin (5400+ ) (ii) csc (-9000+ ) (iii) tan ( - 3600) (iv) cos (- 4500- )

Solution

  1. sin (5400+ ) = sin ( 6 90O 0+ ) = - sin .....[n is even and angle is in Quad III]

  2. csc (- 9000+ ) = csc (10 900 + ) = - csc [csc (-x) = -csc x, n is even and angle is in Quad. II]

  3. tan (- 3600 ) = tan (- 4 900 + ) = - (-tan ) [tan (-x) = -tan x, n is even and angle is in Quad.IV]

  4. cos (- 4500 - ) = cos ( - 5 900- ) = - sin [cos (- ) = cos , n is odd and angle is in Quad (I)]

Example 4

Evaluate tan (-9450) and sin (46200)

Solution

  1. tan (-9450) = - tan 9450 .....[tan (- ) = - tan ]

    = - tan (10 90 0+ 450 )

    = - tan 450 .....[n is even and angle is in Quad III ]

    = -1

  2. sin 4620 0 = sin ( 51 900+ 300)

    = -cos 300 .....( n is even and Quadrant IV)

Example 5 Prove that tan 4950cot (- 4050) + cot ( 4950) tan (- 5850 ) = 2

Solution

L.H.S = tan 4950 cot (- 4050) + cot ( 4950) tan ( -5850)

= tan 4950 (- cot 4050) + cot 4950(- tan 5850)     .....[cot (- ) = - cot and tan (- ) = -tan ]

= - tan (5 900+ 450) cot (4 900 + 450) - cot ( 5 900+ 450) tan ( 6 900 + 450) = - (- cot 450) (cot 450) - (- tan 450) (tan 450)

= 1 1 + 1 1     .....[cot 450 = tan 450= 1]

= 2

[next page]

Index

Trignometric Identities

4.1 Fundamental Identities
4.2 The addition formulas
4.3 The multiple-angle (Double & Half angle) formulas
4.4 Tangent Identities
4.5 Factorization & Defactorization

Supplementary Problems


Chapter 5

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