Exercise 9.2
Prove that: \[\mathbf{sin}\text{ }\mathbf{5x}\text{ }=\text{ }\mathbf{5}\text{ }\mathbf{sin}\text{ }\mathbf{x}\text{ }\text{ }\mathbf{20}\text{ }\mathbf{si}{{\mathbf{n}}^{\mathbf{3}}}~\mathbf{x}\text{ }+\text{ }\mathbf{16}\text{ }\mathbf{si}{{\mathbf{n}}^{\mathbf{5}}}~\mathbf{x}\]
Let us consider LHS: \[sin\text{ }5x\] We can write \[sin\text{ }5x\] as, \[sin\text{ }5x\text{ }=\text{ }sin\text{ }\left( 3x\text{ }+\text{ }2x \right)\] From formula, \[Sin\text{ }\left( x\text{...
Prove that: \[\mathbf{sin}\text{ }\mathbf{5x}\text{ }=\text{ }\mathbf{5}\text{ }\mathbf{co}{{\mathbf{s}}^{\mathbf{4}}}~\mathbf{x}\text{ }\mathbf{sin}\text{ }\mathbf{x}\text{ }\text{ }\mathbf{10}\text{ }\mathbf{co}{{\mathbf{s}}^{\mathbf{2}}}~\mathbf{x}\text{ }\mathbf{si}{{\mathbf{n}}^{\mathbf{3}}}~\mathbf{x}\text{ }+\text{ }\mathbf{si}{{\mathbf{n}}^{\mathbf{5}}}~\mathbf{x}\]
Let us consider LHS: \[sin\text{ }5x\] We can write \[sin\text{ }5x\] as, \[sin\text{ }5x\text{ }=\text{ }sin\text{ }\left( 3x\text{ }+\text{ }2x \right)\] From formula, \[Sin\text{ }\left( x\text{...
Prove that: \[\mathbf{co}{{\mathbf{s}}^{\mathbf{3}}}~\mathbf{x}\text{ }\mathbf{sin}\text{ }\mathbf{3x}\text{ }+\text{ }\mathbf{si}{{\mathbf{n}}^{\mathbf{3}}}~\mathbf{x}\text{ }\mathbf{cos}\text{ }\mathbf{3x}\text{ }=\text{ }\mathbf{3}/\mathbf{4}\text{ }\mathbf{sin}\text{ }\mathbf{4x}\]
We know that, \[cos\text{ }3\theta \text{ }=\text{ }4co{{s}^{3}}\theta \text{ }\text{ }3cos\theta \] SO, \[4\text{ }co{{s}^{3}}\theta \text{ }=\text{ }cos3\theta \text{ }+\text{ }3cos\theta \]...
Prove that: \[\mathbf{4}\text{ }(\mathbf{co}{{\mathbf{s}}^{\mathbf{3}}}~\mathbf{1}{{\mathbf{0}}^{\mathbf{o}}}~+\text{ }\mathbf{si}{{\mathbf{n}}^{\mathbf{3}}}~\mathbf{2}{{\mathbf{0}}^{\mathbf{o}}})\text{ }=\text{ }\mathbf{3}\text{ }(\mathbf{cos}\text{ }\mathbf{1}{{\mathbf{0}}^{\mathbf{o}}}~+\text{ }\mathbf{sin}\text{ }\mathbf{2}{{\mathbf{0}}^{\mathbf{o}}})\]
Take LHS: \[4\text{ }(co{{s}^{3}}~{{10}^{o}}~+\text{ }si{{n}^{3}}~{{20}^{o}})\] We know that, \[sin\text{ }{{60}^{o}}~=~\sqrt{3/2}\text{ }=\text{ }cos\text{ }{{30}^{o}}\] \[Sin\text{...
Prove that: \[\mathbf{sin}\text{ }\mathbf{5x}\text{ }=\text{ }\mathbf{5}\text{ }\mathbf{sin}\text{ }\mathbf{x}\text{ }\text{ }\mathbf{20}\text{ }\mathbf{si}{{\mathbf{n}}^{\mathbf{3}}}~\mathbf{x}\text{ }+\text{ }\mathbf{16}\text{ }\mathbf{si}{{\mathbf{n}}^{\mathbf{5}}}~\mathbf{x}\]
Take LHS: \[sin\text{ }5x\] We can write \[sin\text{ }5x\] as, \[sin\text{ }5x\text{ }=\text{ }sin\text{ }\left( 3x\text{ }+\text{ }2x \right)\] From formula, \[Sin\text{ }\left( x\text{ }+\text{ }y...