# How do you find the smallest angle in a right angled triangle whose side lengths are 6cm, 13cm and 14 cm?

Jun 10, 2016

The smallest angle is ${25.34}^{o}$

#### Explanation:

Since the smallest angle is always opposite the shortest side, we will solve for the value of angle A using the Law of Cosine

${a}^{2} = {b}^{2} + {c}^{2} - 2 b c C o s A$

$a = 6$
$b = 13$
$c = 14$

${6}^{2} = {13}^{2} + {14}^{2} - 2 \left(13\right) \left(14\right) C o s A$

$36 = 169 + 196 - 2 \left(13\right) \left(14\right) C o s A$

$36 = 365 - 364 C o s A$

$36 - 365 = \cancel{365} \cancel{- 365} - 364 C o s A$

$- 329 = - 364 C o s A$

$\frac{- 329}{- 364} = \frac{\cancel{- 364} C o s A}{\cancel{- 364}}$

$0.9038 = C o s A$

$C o {s}^{-} 1 \left(0.9038\right) = A$

$A = {25.34}^{o}$

Jul 30, 2016

$\cos A = 0.9 .38$

A = 25.33°

#### Explanation:

Sketch taken from Brian M's answer.

The Cosine Rule can be written in two ways: one for finding a side, and the transposed formula for finding an angle.

It Really is worth learning both Forms!

For side BC: ${a}^{2} = {b}^{2} + {c}^{2} - 2 b c C o s A$

For angle A: $C o s A = \frac{{b}^{2} + {c}^{2} - {a}^{2}}{2 b c}$

The smallest angle is always opposite the shortest side, so we need to find the size of Angle A in this case.

Substitute the given values:

$C o s A = \frac{{13}^{2} + {14}^{2} - {6}^{2}}{2 \times 13 \times 14}$

There is little point in doing long-hand calculations, or even calculating the individual answers. It it the final value that we need.
Use a calculator, making sure to divide the whole of the numerator by the whole of the denominator. (Use brackets to ensure this.)

$C o s A = \frac{\left({13}^{2} + {14}^{2} - {6}^{2}\right)}{\left(2 \times 13 \times 14\right)}$

$\cos A = 0.9 .38$

A = 25.33°