Multiple angles

A trigonometric identity  is an equality between expressions that involve trigonometric functions and that is true for all angle values (multiple angles) in which they are defined.

Multiple angle identities

Trigonometric identities of the double angle

\ large \ begin {align *} \ bullet & \ hspace {1.0em} \ sin 2a = 2 \ sin a \ cdot \ cos a \\ \ bullet & \ hspace {1.0em} \ cos 2a = 2 \ cos ^ { 2} a - \ sin ^ {2} a \\ \ bullet & \ hspace {1.0em} \ tan 2a = \ frac {2 \ tan a} {1- \ tan ^ {2} a} \\ \ end { align *}

Trigonometric identities of the angle triple

Trigonometric identities of the half angle

Demonstration

Double angle

sin 2╬▒

In this case, the simplest is to use the identities of the sum of two angles:

doing ╬▓ = ╬▒, we have:

cos 2╬▒

For the cosine of the double angle we repeat the same procedure above:

Considering that:

and making ╬▓ = ╬▒:

tg 2╬▒

How:

So:

Triple angle

To demonstrate the sine and cosine of the triple angle we can do it in two ways. The first is following a procedure similar to the previous one when considering that:

We develop the sum of angles and then the double angle.

Another way is by using the De Moivre's formula, which establishes that for any complex number (as well as for any real) "x" and an integer "n", it is verified that:

Where i is the imaginary number:

sin 3╬▒ and cos 3╬▒

Doing n = 3 and x = ╬▒ in Moivre's formula, we have:

As i┬▓ = -1 and i┬│ = -i:

Equalizing the real and the imaginary part we obtain:

Considering that:

we have:

tg 3╬▒