Sin(x+y+z) Identity, Formula, Proof

The sin(x+y+z) identity is given by sin(x+y+z) = sinx cosy cosz + cosx siny cosz + cosx cosy sinz – sinx siny sinz. In this post, we will establish the formula of sin(x+y+z).

Formula of sin(x+y+z)

The formula of sin(x+y+z) is given as follows:

$\sin(x+y+z) = \sin x \cos y \cos z + \cos x \sin y \cos z$ $+ \cos x \cos y \sin z – \sin x \sin y \sin z$.

Proof:

To establish the above formula of sin(x+y+z), we will use the following two formulas:

  1. sin(a+b) = sina cosb + cosa sinb
  2. cos(a+b) = cosa cosb – sina sinb

Now, sin(x+y+z)

= sin ((x+y)+z)

= sin(x+y) cosz + cos(x+y) sinz by the above formula 1 with a=x+y and b=z.

= [sinx cosy + cosx siny] cosz + [cosx cosy – sinx siny] sinz, using the above two formuals.

= sinx cosy cosz + cosx siny cosz + cosx cosy sinz – sinx siny sinz.

So the formula of sin(x+y+z) formula is equal to sin(x+y+z) = sinx cosy cosz + cosx siny cosz + cosx cosy sinz – sinx siny sinz. This is proved by applying the formulas of sin(a+b) and cos(a+b).

More Trigonometric Formulas:

cos(a+b) cos(a-b) Formula

sin(a+b) sin(a-b) Formula

cosec2x Formula | sec2x Formula

cot2x Formula | cosx cosy Formula

FAQs

Q1: What is the Formula of sin(x+y+z)?

Answer: The formula of sin(x+y+z) is sinx cosy cosz + cosx siny cosz + cosx cosy sinz – sinx siny sinz.

Q2: What is the Formula of sin(a+b+c)?

Answer: The formula of sin(a+b+c) is sina cosb cosc + cosa sinb cosc + cosa cosb sinc – sina sinb sinc.

Spread the love