Derivation of trigonometric Triple angle identities
Triple angle identities give relationship between trigonometric functions of an angle and the one that is three multiples of the angle. Triple angle identities can also expressed to transform an trig function of power of three to trigonometric functions of lower power.
There are 6 triple angle identities corresponding to 6 elementary trigonometric functions. However, the most frequent used ones are for sines, cosine and tangent functions. In order to derive the triple angle identities, it is the premise to recognize the fundamental trigonometric identities such as double angle, sum and product identities.
Triple angle identity for sine function
First, convert the triple angle to the sum of double angle and single angle. And then, use the Sum identity for sine function, double angle identities and Pythagorean identity for derivation.
Hence, triple angle identity for sine function looks a form of a depressed cubic function.
Noticed the special numbers of coefficients of terms on right hand side, the triple angle identity could be transformed to the interesting form.
The triple identity could be transformed to the form of product of sine function of single angle and two symmetric angles about 60°.
Triple angle identity for cosine function
Proof of triple angle identity for cosine function is similar to the steps for sine function.
Hence, triple angle identity for cosine function is a form of a depressed cubic function in terms of cosine of single angle.
In the same manner, the triple function for cosine function could be written as the product of cosine of single angle and two symmetric angles.
Rewriting equation (1) and (3) gives the form of power reducing
for sine function
for cosine function
Triple angle identity for tangent function
Triple angle identity for tangent function could be obtained by dividing sine by cosine identity in their product form.
Dividing (2) by (4) yields
Using the sum and difference identities for tangent functions
\tan3α could be represented as the expression in terms of \tan α.