Double Angle Trig Identities

Introduction for double angle trig identities:

In mathematics, double angle trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables. Geometrically, these are identities involving certain function of one or more angles. These are distinct from the triangle identities, which are identities involving both angles and side lengths of a triangle. These identities are useful whenever expression involving trigonometric functions need to be simplified. (Source.Wikipedia)

Double Angle Trig Identities:

sin2B = 2 sinB. cosB

cos2B = cos2B - sin2B

cos2B = 1 - 2sin2B

cos2B = 2 cos2B - 1

tan2B ='(2tanB) / (1 - tan^2B) '

sin2B ='(2 tanB) /(1 + tan^2 B)'

cos2B ='[1 - tan^2B] /[1 + tan^2B]'

Double Angle Trig Identities Examples:

Example 1:

Prove that cos4C - sin4C = cos2C

Solution:

LHS. = (cos2C + sin2C) (cos2C - sin2C)

= 1 . cos2C

= cos2C

= RHS.

Example 2:

Show that 4 sinB sin (60 + B) . sin (60 - B) = sin3B

Solution:

LHS. = 4 sinB sin (60 + B) . sin (60 - B)

= 4sinB { sin (60 + B) . sin (60 - B)}

= 4sinB {sin260 - sin2B}

= 4sinB{'3/2 ' - sin2B} = 3sinB - 4sin3B = sin3B

= RHS.

Example 3:

If tanA ='[1 -cosG ]/sinG ' , prove that tan2G = tanD, where G and D are acute angles.

Solution:

RHS ='[1 - cosG ]/sinG' =2sin2' G/2' / 2sin'G/2' . cos'G/2'

=sin'G/2 ' cos'G/2'

= tan'G/2'

? tanG/2 = tanA

? A ='G/2 ' ? G = 2A

Therefore tan2G = tanB

Example 4:

Prove that 4cos20 cos40 cos80 ='1/2'

Solution:

LHS. = 4cos20 cos40 cos80

= 4cos20 {cos (60 - 20) cos(60 + 20)}

= 4cos20 [cos260 - sin220]

= 4cos20{'1/4 ' - sin220}

= cos20 {1 - 4(1 - cos 220)}

= {4cos320 - 3 cos20} = [cos3 20]

=1 cos60 ='1/2' = RHS.

Practice Problem for Double Angle Trig Identities:

1 Prove that

(i) cos2 {p/4 - A} - sin2{p/4 - A} = sin2A (ii) sec2A+tan2A = tan p/4 + A

2. Prove the following:

(i) sin15 cos15 =1/4 (ii) 2sinp/8cosp/8 =v2

3.Prove that[1 + sin a - cos a]/[1 + sin a+ cos a] = tana/2

4.Show that cos3 p/9 - 6cosp/9 = 1/8

by: jeri

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