How do you express $\frac{sin 4 θ}{sin θ}$ $(sin 4 theta)/(sin theta)$ in terms of $sin θ$ $sin theta$ and $cos θ$ $cos theta$ ?

Question

How do you express $\frac{sin 4 θ}{sin θ}$ $(sin 4 theta)/(sin theta)$ in terms of $sin θ$ $sin theta$ and $cos θ$ $cos theta$ ?

1 Answer

Impact of this question

3758 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-03-11T08:41:26

$\frac{sin 4 θ}{sin θ} = 4 {cos}^{3} θ - 4 cos θ {sin}^{2} θ = 4 cos θ ({cos}^{2} θ - {sin}^{2} θ)$ $(sin 4 theta) / (sin theta) = 4 cos^3 theta - 4 cos theta sin^2 theta = 4 cos theta(cos^2 theta - sin^2 theta)$

Explanation:

By de Moivre's formula we have:

$cos n θ + i sin n θ = {(cos θ + i sin θ)}^{n}$ $cos n theta + i sin n theta = (cos theta + i sin theta)^n$

So:

$cos 4 θ + i sin 4 θ$ $cos 4 theta + i sin 4 theta$

$= {(cos θ + i sin θ)}^{4}$ $= (cos theta + i sin theta)^4$

$= ((4),(0)) cos^4 theta + ((4),(1)) i cos^3 theta sin theta + ((4),(2)) i^2 cos^2 theta sin^2 theta + ((4),(3)) i^3 cos theta sin^3 theta + ((4),(4)) i^4 sin^4 theta$

$= cos^4 theta + 4 i cos^3 theta sin theta -6 cos^2 theta sin^2 theta -4i cos theta sin^3 theta + sin^4 theta$

$= (cos^4 theta -6 cos^2 theta sin^2 theta + sin^4 theta) + i(4 cos^3 theta sin theta -4 cos theta sin^3 theta)$

Equating imaginary parts, we find:

$sin 4 theta = 4 cos^3 theta sin theta - 4 cos theta sin^3 theta$

So:

$(sin 4 theta) / (sin theta) = 4 cos^3 theta - 4 cos theta sin^2 theta$

$color(white)((sin 4 theta) / (sin theta)) = 4 cos theta(cos^2 theta - sin^2 theta)$

Science

Math

Humanities

... and beyond

How do you express $\frac{sin 4 θ}{sin θ}$ $(sin 4 theta)/(sin theta)$ in terms of $sin θ$ $sin theta$ and $cos θ$ $cos theta$ ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you express (sin 4 theta)/(sin theta)sin4θsinθ(sin 4 theta)/(sin theta) in terms of sin thetasinθsin theta and cos thetacosθcos theta ?

1 Answer

Explanation:

Related questions

Impact of this question

How do you express $\frac{sin 4 θ}{sin θ}$ $(sin 4 theta)/(sin theta)$ in terms of $sin θ$ $sin theta$ and $cos θ$ $cos theta$ ?