subject: Double Angle Trig Identities [print this page] 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.
