Cosine rule question (1 Viewer)

kpad5991 · Jan 16, 2017

The sides of a triangle are in the ratio 5:16:19. Find the smallest and largest angles of the triangle

InteGrand · Jan 16, 2017

kpad5991 said:
The sides of a triangle are in the ratio 5:16:19. Find the smallest and largest angles of the triangle

$\noindent Let $\alpha$ be the smallest angle and $\beta$ the largest. The smallest angle is the one opposite the shortest side and the largest angle is the one opposite the longest side. So by the cosine rule, we have$

$\begin{align*}\cos \alpha &= \frac{16^{2} + 19^{2} -5^{2}}{2\times 16\times 19} \\ &= \frac{37}{38}\\ \Rightarrow \alpha &= \cos^{-1}\left(\frac{37}{38}\right).\end{align*}$

$\noindent Also,$

$\begin{align*}\cos \beta &= \frac{5^{2} + 16^{2} -19^{2}}{2\times 5\times 16} \\ &= - \frac{1}{2}\\ \Rightarrow \beta &= \frac{2\pi}{3} = 120^{\circ}.\end{align*}$

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Cosine rule question (1 Viewer)

kpad5991

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InteGrand

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