Trigonometry
Trigonometric functions of double angles, double angle formulas
Trigonometric functions expressed by the half angle
Trigonometric identities, examples
Trigonometric functions of double angles, double angle formulas
By substituting b with a in the sum formulas,
sin (a + b) = sin a · cos b + cosa · sin b      and     cos (a + b) = cos a · cos b - sin a · sin b,
 and
thus for example,   sin 2a = sin (a + a) = sin a · cos a + cosa · sin a = 2sin a cos a   so we get,
 sin 2a = 2 sin a cos a and cos 2a = cos2 a - sin2 a
 and
The double angle formula for the cosine function can be expressed by sine or cosine function using the
identity sin2 a + cos2 a = 1,
 cos 2a = 2cos2 a - 1 or cos 2a = 1 - 2sin2 a
 and 1 + cos 2a = 2 cos2 a or 1 - cos 2a = 2sin2 a
Trigonometric functions expressed by the half angle
Substituting a/2 in the double angle formulas we obtain trigonometric functions expressed by the half angle,
 and
 and
and
 or
 or
Trigonometric identities examples
Example:   Using known values, sin 60° = Ö3/2 and sin 45° = Ö2/2 evaluate sin 105°.
Solution:  Applying the sum formula for the sine function,  sin (a + b) = sin a · cos b + cosa · sin b
therefore,   sin 105° = sin (60° + 45°) = sin 60° · cos 45° + cos60° · sin 45°
Example:   Express tan 3x in terms of tan x.
Solution:   Using the sum formula and the double angle formula for the tangent function,
 Example:   Prove the identity
 Solution:
 Example:   Prove the identity
 Solution:
Example:   If  tan a = 3/4, find tan a/2.
 Solution:   Use formula to express tan a/2  in terms of  tan a.
 Example:   Prove the identity
 Solution:   Substitute then
Example:   Express the given difference  sin 61° - sin 59°  as a product.
 Solution:   Since
Example:   Prove the identity  sin a + sin (a + 120°) + sin (a + 240°) = 0.
Solution:   Applying the sum formula to the last two terms on the left side of the identity we get,
 Example:   Prove the identity
 Solution:   Using the formula for the sum of the tangent
Trigonometry contents A