Trigonometry
Trigonometric functions expressed by the tangent of the half angle
Half angle formulas
Trigonometric functions expressed by the cosine of the double angle
Trigonometric identities, examples
Trigonometric functions expressed by the tangent of the half angle
Replacing a by a/2 in the above identities, we get
Half angle formulas
Using the identities in which trigonometric functions are expressed by the half angle,
 => =>
and applying the definitions of the functions, tangent and cotangent
 => =>
Trigonometric functions expressed by the cosine of the double angle
Replacing a/2 by a in the above identities, we get
Trigonometric identities, examples
Example:   Express sin 3x in terms of sin x.
Solution:   Using the sum formula and the double angle formula for the sine function,
sin 3x = sin (2x + x) = sin 2x · cos x + cos 2x · sin x = 2sin x cos x · cos x + (cos2 x - sin2 x) · sin x
= 2sin x · (1 - sin2 x) + (1 - 2sin2 x) · sin x = 3sin x - 4sin3 x.
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:   Prove the identity
 Solution:   Using the formula for the sum of the tangent
 Example:   Prove that
Solution:  Replace sin a by cos (p/2 - a) and cos a by sin (p/2 - a) and use the sum to product formula
Pre-calculus contents F