Question

Prove that sin3theta = 3sintheta - 4sin³theta​

Answer 1

$\large\underline{\sf{Solution-}}$

Consider,

$\rm \: sin3\theta$

$\rm \: = \: sin(2\theta + \theta )$

$\rm \: = \: sin2\theta \: cos\theta + sin\theta \: cos2\theta$

$\rm \: = \: (2sin\theta cos\theta )cos\theta + sin\theta (1 - {2sin}^{2}\theta )$

$\rm \: = \: 2sin\theta {cos}^{2}\theta + sin\theta - 2 {sin}^{3}\theta$

$\rm \: = \: 2sin\theta(1 - {sin}^{2}\theta) + sin\theta - 2 {sin}^{3}\theta$

$\rm \: = \: 2sin\theta - 2{sin}^{3}\theta + sin\theta - 2 {sin}^{3}\theta$

$\rm \: = \: 3sin\theta - 4{sin}^{3}\theta$

Hence,

$\rm \:\rm\implies \:\boxed{ \rm{ \:sin3\theta = \: 3sin\theta - 4{sin}^{3}\theta \: \: }}$

$\rule{190pt}{2pt}$

Formulae Used :-

$\boxed{ \rm{ \:sin(x + y) = sinx \: cosy \: + \: siny \: cosx \: }}$

$\boxed{ \rm{ \:sin2x = 2sinxcosx \: }}$

$\boxed{ \rm{ \:cos2x = 1 - {2sin}^{2}x \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered} { \boxed{ \begin{array}{c} \underline{\underline{ \color{orange} \text{Additional \: lnformation}}} \\ \\ & \rm \: sin2x \: = 2 \: sinx \: cosx\:\\ & \rm \: cos2x = 1 - {2sin}^{2}x \\ & \rm \: cos2x = {2cos}^{2}x - 1 \\ & \rm \: cos2x = {cos}^{2}x - {sin}^{2}x \\ & \rm \:tan2x = \dfrac{2tanx}{1 - {tan}^{2} x} \\ & \rm \: sin2x = \frac{2tanx}{1 + {tan}^{2}x } \\ & \rm \:sin3x = 3sinx - {4sin}^{3}x \\ & \rm \: cos3x = {4cos}^{3}x - 3cosx \\ & \rm \: tan3x = \dfrac{3tanx - {tan}^{3} x}{1 - {3tan}^{2}x} \end{array}}}\end{gathered}$

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: Formulae}}}} \\ \\ \bigstar \: \bf{sin(x - y) = sinx \: cosy \: - \: siny \: cosx}\\ \\ \bigstar \: \bf{sin(x + y) = sinx \: cosy \: + \: siny \: cosx}\\ \\ \bigstar \: \bf{cos(x + y) = cosx \: cosy \: - \: sinx \: siny}\\ \\ \bigstar \: \bf{cos(x - y) = cosx \: cosy \:+\: siny \: sinx}\\ \\ \bigstar \: \bf{tan(x + y) = \dfrac{tanx + tany}{1 - tanx \: tany} }\\ \\ \bigstar \: \bf{tan(x - y) = \dfrac{tanx - tany}{1 + tanx \: tany} }\\ \\ \end{array} }}\end{gathered}\end{gathered}\end{gathered}$