The sides of a triangle are 7- 4-3- and -13 yards respectively- Find the number of degrees in its smallest angle-

Let a=7, b=4 #8730;3 and c= #8730;13.In a triangle, the smallest angle has shortest opposite side.c= #8730;13 is shortest side.by using cosine formulaeSub ...

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Standard XII

Chemistry

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Question

The sides of a triangle are $7, 4 \sqrt{3}$ , and $\sqrt{13}$ yards respectively. Find the number of degrees in its smallest angle.

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Solution

Let a=7, b=4√3 and c=√13.
In a triangle, the smallest angle has shortest opposite side.
c=√13 is shortest side.
by using cosine formulae
$\cos C=\frac{a^2+b^2-c^2}{2ab}$
Substitute the given value in the above formula.
$\cos C=\frac{(7)^2+(4\sqrt{3})^2-(\sqrt{13})^2}{2(7)(4\sqrt{3})}$
$\cos C=\frac{49+48-13}{56\sqrt{3}}$
$\cos C=\frac{84}{56\sqrt{3}}$
$\cos C=\frac{3}{2\sqrt{3}}$
On rationalization.
$\cos C=\frac{3\times \sqrt{3}}{2\sqrt{3}\times \sqrt{3}}$
$\cos C=\frac{3\times \sqrt{3}}{2\times 3}$
$\cos C=\frac{\sqrt{3}}{2}$
$\cos C=\cos 30^{\circ}$
On comparing both sides.
$C=30^{\circ}$

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