This is the question:
Question (e): By considering the solutions of the equation \(\displaystyle \cos\left(5 \theta \right) \, =\, 0,\)
show that \(\displaystyle \cos\left(\dfrac{\pi}{10}\right)\, =\, \sqrt{\dfrac{5\, +\, \sqrt{5\,}}{8}\,}\) and state the value of \(\displaystyle \cos\left(\dfrac{7 \pi}{10}\right)\)
This is the solution:
Consider: \(\displaystyle 16\, \cos^5\left(\theta\right)\, -\, 20\, \cos^3\left(\theta\right)\, +\, 5\, \cos\left(\theta\right)\, =\, 0\)
\(\displaystyle \cos\left(\theta\right)\, \left[16\cos^4\left(\theta\right)\, -\, 20\, \cos^2\left(\theta\right)\, +\, 5\right]\, =\, 0\)
\(\displaystyle \cos^2\left(\theta\right)\, =\, \dfrac{20\, \pm\, \sqrt{400\, -\, 4(16)(5)\,}}{32},\,\) \(\displaystyle \cos\left(\theta\right)\, =\, 0\)
\(\displaystyle \cos\left(\theta\right)\, =\, \pm\, \sqrt{\dfrac{20\, \pm\, \sqrt{400\, -\, 4(16)(5)\,}}{32}\,}\)
\(\displaystyle \cos\left(\dfrac{\pi}{10}\right)\, =\, \pm\, \sqrt{\dfrac{20\, \pm\, \sqrt{400\, -\, 4(16)(5)\,}}{32}\,}\) since max value of cosine \(\displaystyle \Rightarrow\) angle closest to zero
\(\displaystyle \cos\left(\dfrac{\pi}{10}\right)\, =\, \sqrt{\dfrac{4.5\, +\,4\, \sqrt{25\, -\, 4(5)\,}}{4.8}\,}\, =\, \sqrt{\dfrac{5\, +\, \sqrt{5\,}}{8}\,}\)
\(\displaystyle \cos\left(\dfrac{7 \pi}{10}\right)\, =\, -\sqrt{\dfrac{5\, -\, \sqrt{5\,}}{8}\,}\)
Could please explain in details what is going on in the last 3 lines of this solution
Question (e): By considering the solutions of the equation \(\displaystyle \cos\left(5 \theta \right) \, =\, 0,\)
show that \(\displaystyle \cos\left(\dfrac{\pi}{10}\right)\, =\, \sqrt{\dfrac{5\, +\, \sqrt{5\,}}{8}\,}\) and state the value of \(\displaystyle \cos\left(\dfrac{7 \pi}{10}\right)\)
This is the solution:
Consider: \(\displaystyle 16\, \cos^5\left(\theta\right)\, -\, 20\, \cos^3\left(\theta\right)\, +\, 5\, \cos\left(\theta\right)\, =\, 0\)
\(\displaystyle \cos\left(\theta\right)\, \left[16\cos^4\left(\theta\right)\, -\, 20\, \cos^2\left(\theta\right)\, +\, 5\right]\, =\, 0\)
\(\displaystyle \cos^2\left(\theta\right)\, =\, \dfrac{20\, \pm\, \sqrt{400\, -\, 4(16)(5)\,}}{32},\,\) \(\displaystyle \cos\left(\theta\right)\, =\, 0\)
\(\displaystyle \cos\left(\theta\right)\, =\, \pm\, \sqrt{\dfrac{20\, \pm\, \sqrt{400\, -\, 4(16)(5)\,}}{32}\,}\)
\(\displaystyle \cos\left(\dfrac{\pi}{10}\right)\, =\, \pm\, \sqrt{\dfrac{20\, \pm\, \sqrt{400\, -\, 4(16)(5)\,}}{32}\,}\) since max value of cosine \(\displaystyle \Rightarrow\) angle closest to zero
\(\displaystyle \cos\left(\dfrac{\pi}{10}\right)\, =\, \sqrt{\dfrac{4.5\, +\,4\, \sqrt{25\, -\, 4(5)\,}}{4.8}\,}\, =\, \sqrt{\dfrac{5\, +\, \sqrt{5\,}}{8}\,}\)
\(\displaystyle \cos\left(\dfrac{7 \pi}{10}\right)\, =\, -\sqrt{\dfrac{5\, -\, \sqrt{5\,}}{8}\,}\)
Could please explain in details what is going on in the last 3 lines of this solution
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