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Example 6 Prove that

Solution

Example 7 Write sin 2340 in the form cos b, 0 < b< 900

Solution sin 2340 = sin ( 1800 + 540 )

= - sin 54 0 . . . . [sin (18 0 + ) = - sin ]

= - sin (900 - 540) . . . . [sin = cos (900- )]

= - cos 360

Example 8 Show that cos b cos (a - b) - sin b sin ( a - b ) = cos a

Solution

L.H.S = cos b cos ( a - b ) - sin b sin ( a - b )

= cos [ b + ( a - b ) ]

= cos a

= R.H.S.

Example 9

Solution

Example 10

Prove that

Solution

Example 11

Prove that

Solution

Example 12

If tan 250 = a

Solution:

Example 13 Prove that

Solution:

EXAMPLE 14 Prove that sin 190+ cos 190 = tan (640)

cos 190 - sin 190

Solution :

EXAMPLE 15 Find sin (a + b) if sin a = - 4/5, cos b = 15/ 17 and a and b are in Quadrant IV. Also state the Quadrant of (a + b).

Solution

Example 16

If sin A = 5/13 and cos B = - 4 / 5. Find cos (A + B). State the Quadrant of (A + B) where A and B are obtuse angles.

Solution

Since cosine ratio is +ve, the terminal arm of (A+ B) lies in Quad IV.

Double - Angle

From the sum and difference formulas for sine and cosine, one can get `double angle' formulas as :-

  1. sin 2q = sin (q + q)
    = sin q cos q + cos q sin q
    = 2 sin q cos q

  2. cos 2q = cos (q + q)
    = cos q cos q - sin q sin q
    = cos2q - sin2q      .....(1)


    using Pythagorean identity sin2 q = 1- cos2 q in (1)
    \ cos 2 q = cos2 q - (1- cos2q)
    = 2cos2q - 1

    Also using cos2q = 1 - sin2q in (1)
    cos 2q = 1 - sin 2 q - sin2q
    = 1 - 2 sin2q

Trigonometric Ratios of (3a)

  1. sin 3a = sin (2a + a)
    = sin 2a cos a + cos 2a sin a
    = (2 sin a cos a) cos a + (1 - 2sin2 a) sin a
    = 2 sin a cos2a + sin a - 2 sin3a
    = 2 sin a (1- sin 2 a) + sin a - 2sin3a
    = 2 sin a - 2 sin 3 a + sin a - 2 sin3a
    = 3 sin a - 4 sin 3 a

  2. cos 3a = cos (2a + a)
    = cos 2a cos a - sin 2a sin a
    = (2 cos2 a - 1) cos a - (2 sin a cos a) sin a
    = 2 cos3 a - cos a - 2 sin2a cos a
    = 2 cos3 a - cos a - 2 (1 - cos2a) cos a
    = 2 cos3a - cos a - 2 cos a + 2 cos3a
    = 4 cos3a - 3 cos a

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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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