Question

solve \\( \sin ^ { 2 } ( w ) = - 8 \cos ( w ) \\) for all solutions \\( 0 \leq w < 2 \pi \\).

\\( w = \\)

give your answers as values accurate to at least two decimal places in a list separated by commas.

question help: \\( \square \\) video 1 \\( \square \\) video 2

Explanation:

Step1: Use the Pythagorean identity

We know that \(\sin^{2}(w)=1 - \cos^{2}(w)\). Substitute this into the given equation \(\sin^{2}(w)=- 8\cos(w)\), we get \(1-\cos^{2}(w)=-8\cos(w)\).
Rearrange it to the quadratic form: \(\cos^{2}(w)-8\cos(w)-1 = 0\).
Let \(x = \cos(w)\), then the equation is \(x^{2}-8x - 1=0\).

Step2: Solve the quadratic equation

Use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for the quadratic equation \(ax^{2}+bx + c = 0\). Here \(a = 1\), \(b=-8\), \(c=-1\).
\(x=\frac{8\pm\sqrt{(-8)^{2}-4\times1\times(-1)}}{2\times1}=\frac{8\pm\sqrt{64 + 4}}{2}=\frac{8\pm\sqrt{68}}{2}=\frac{8\pm2\sqrt{17}}{2}=4\pm\sqrt{17}\).
Since \(-1\leqslant\cos(w)\leqslant1\), \(x = 4+\sqrt{17}\approx4 + 4.123=8.123\) (rejected) and \(x=4-\sqrt{17}\approx4-4.123=-0.123\).
So \(\cos(w)=4-\sqrt{17}\approx - 0.123\).

Step3: Find the values of \(w\)

If \(\cos(w)=a\), then \(w=\cos^{-1}(a)+2k\pi\) or \(w = 2\pi-\cos^{-1}(a)+2k\pi\), \(k\in\mathbb{Z}\).
For \(0\leqslant w<2\pi\), \(w=\cos^{-1}(-0.123)\approx1.69\) and \(w = 2\pi-\cos^{-1}(-0.123)\approx4.59\).

Answer:

\(1.69,4.59\)