By using DeMoivre's theorm express
$$\frac{\sin 7\theta}{\sin \theta}$$
in the powers of Sine only
answer given in the book is
$$7-56\sin ^2\theta+112\sin ^4 \theta-64\sin^6 \theta$$
can any one help to solve the question
[Math] Express $\frac{\sin 7\theta}{\sin \theta}$ in powers of $\sin \theta$ only
complex numberstrigonometry
Related Question
- Trigonometry – Expressing cos(?) – sqrt(3)sin(?) = r sin(? – ?)
- [Math] Solve: $-\frac{1}{\sqrt{2}} < \sin \theta + \cos \theta < \frac{1}{\sqrt{2}}$
- [Math] Use De Moivre’s Theorem to express $\sin(3\theta)$ in terms of the powers of $\sin (\theta)$ and $\cos(\theta)$
- Solve for $\theta$ in the equation $\sin(2\theta) = \sin(\theta)$ using complex numbers
Best Answer
Steps To Carry Out
1) First take a look at this link which is a guide for DeMoivre's formula.
2) Using step 1 show that
$$\sin (7x) = 64\sin \left( x \right)\cos {\left( x \right)^6} - 80\sin \left( x \right)\cos {\left( x \right)^4} + 24\sin \left( x \right)\cos {\left( x \right)^2} - \sin \left( x \right)$$
3) Replace $\cos^2(x)=1-\sin^2(x)$ and obtain
$$\sin (7x) = 7\sin \left( x \right) - 56\sin {\left( x \right)^3} + 112\sin {\left( x \right)^5} - 64\sin {\left( x \right)^7}$$
4) Divide by $\sin(x)$
$${{\sin (7x)} \over {\sin (x)}} = 7 - 56\sin {\left( x \right)^2} + 112\sin {\left( x \right)^4} - 64\sin {\left( x \right)^6}$$