How do you solve the triangle given A=47, a=67, b=83?

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Answer 1 · 2018-05-24T04:54:24

$ˆ A = {34.42}^{\circ}, ˆ B = {53.84}^{\circ}, ˆ C = {91.74}^{\circ}$ $color(purple)(hat A = 34.42^@, hat B = 53.84^@, hat C = 91.74^@$

$a = 47, b = 67, c = 83$ $a = 47, b = 67, c = 83$

As per Law of Cosines,

$a^{2} = b^{2} + c^{2} - 2 b c cos A$ $a^2 = b^2 + c^2 - 2 b c cos A$

$cos A = \frac{b^{2} + c^{2} - a^{2}}{2 b c}$ $cos A = (b^2 + c^2- a^2) / (2bc)$

$cos A = \frac{67^{2} + 83^{2} - 47^{2}}{2 \cdot 67 \cdot 83}$ $cos A = (67^2 + 83^2 - 47^2) / (2 * 67 * 83)$

$cos A = 0.8244$ $cos A = 0.8244$

$ˆ A = {cos}^{- 1} 0.8244 = {34.42}^{\circ}$ $hat A = cos ^-1 0.8244 = 34.42^@$

Similarly, $cos C = \frac{b^{2} + a^{2} - c^{2}}{2 a b}$ $cos C = (b^2 + a^2- c^2) / (2ab)$

$cos C = \frac{47^{2} + 67^{2} - 83^{2}}{2 \cdot 47 \cdot 67}$ $cos C = (47^2 + 67^2 - 83^2) / (2 * 47 * 67)$

$cos C = - 0.0303$ $cos C = -0.0303$

$ˆ C = {91.74}^{\circ}$ $hat C = 91.74^@$

$ˆ B = 180 - ˆ A - ˆ C$ $hat B = 180 - hat A - hat C$

$ˆ B = 180 - 34.42 - 91.74 = {53.84}^{\circ}$ $hat B = 180 - 34.42 - 91.74 = 53.84^@$