Question

$\sin h\left(3z\right)$ equals

• A. $2\sin hz–4\sin h^{3}z$
• B. $4\sin h^{3}z–3\sin hz$
• C. $3\sin hz+4\sin h^{3}z$
• D. None of these

Answer 1

The correct option is C

$3\sin hz+4\sin h^{3}z$

Explanation for the correct answer:

Simplifying the given expression using trigonometric identities:

$$\begin{gathered} \sin h\left(3z\right)=\sin h(2z+z) \\ =\sin h\left(2z\right)\cos hz+\cos h\left(2z\right)\sin hz\left[\because sinh(x+y)=sinhxcoshy+coshxsinhy\right] \\ =(2\sin hz\cos hz)\cos hz+(\cos h^{2}z+\sin h^{2}z)\sin hz\mathbf{}\left[\because sinh\left(2x\right)=2sinhxcoshx\text{and}cosh\left(2x\right)=cos{h}^{2}x+sin{h}^{2}x\right] \\ =2\sin hz\cos h^{2}z+(1+\sin h^{2}z+\sin h^{2}z)\sin hz\left[\because cos{h}^{2}z=1+sin{h}^{2}z\right] \\ =2\sin hz(1+\sin h^{2}z)+(1+2\sin h^{2}z)\sin hz \\ =2\sin hz+2\sin h^{3}z+\sin hz+2\sin h^{3}z \\ =3\sin hz+4\sin h^{3}z \end{gathered}$$

Therefore, option (C) is the correct answer.