Two sides of a triangle are 16.2 cm and 29.6 cm, with the included angle 106◦. What is angle opposite the side 16.2 cm?

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Two sides of a triangle are 16.2 cm and 29.6 cm, with the included angle 106◦. What is angle opposite the side 16.2 cm?

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Answer 1 · 2016-12-02T17:37:01

The angle $A \approx {24.6}^{\circ}$ $A ~~ 24.6^@$

Explanation:

Let side $a = 16.2 cm$ $a = 16.2" cm"$
Let side $b = 29.6 cm$ $b = 29.6" cm"$
Let side $c =$ $c =$ the unknown side.

Let $∠ A =$ $angle A =$ the unknown angle opposite side a.
Let $∠ B =$ $angle B =$ the unknown angle opposite side b.
Let $∠ C = 106^{\circ}$ $angle C = 106^@$

Use the Law of Cosines to find the length of side c:

$c = \sqrt{a^{2} + b^{2} - 2 (a) (c) cos (C)}$ $c = sqrt(a^2 + b^2 - 2(a)(c)cos(C))$

$c = \sqrt{{(16.2 cm)}^{2} + {(29.6 cm)}^{2} - 2 (16.2 cm) (29.6 cm) cos (106^{\circ})}$ $c = sqrt((16.2" cm")^2 + (29.6" cm")^2 - 2(16.2" cm")(29.6" cm")cos(106^@))$

$c \approx 37.46 cm$ $c ~~ 37.46" cm"$

Use the Law of Sines to find $∠ A$ $angle A$

$\frac{sin (A)}{a} = \frac{sin (C)}{c}$ $sin(A)/a = sin(C)/c$

$sin (A) = sin (C) \frac{a}{c}$ $sin(A) = sin(C)a/c$

$A = {sin}^{- 1} (sin (C) \frac{a}{c})$ $A = sin^-1(sin(C)a/c)$

$A = {sin}^{- 1} (sin (106^{\circ}) \frac{16.2 cm}{37.46 cm})$ $A = sin^-1(sin(106^@)(16.2" cm")/(37.46" cm"))$

$A \approx {24.6}^{\circ}$ $A ~~ 24.6^@$

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Two sides of a triangle are 16.2 cm and 29.6 cm, with the included angle 106◦. What is angle opposite the side 16.2 cm?

1 Answer

Explanation:

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