Solve the Triangle a=100 , b=120 , c=85

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Trigonometry

$a = 100$ , $b = 120$ , $c = 85$

Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.

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

Solve the equation.

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

Substitute the known values into the equation.

$A = \arccos (\frac{{(120)}^{2} + {(85)}^{2} - {(100)}^{2}}{2 (120) (85)})$

Simplify the results.

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Simplify the numerator.

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Raise $120$ to the power of $2$ .

$A = \arccos (\frac{14400 + {(85)}^{2} - {(100)}^{2}}{2 \cdot 120 \cdot 85})$

Raise $85$ to the power of $2$ .

$A = \arccos (\frac{14400 + 7225 - {(100)}^{2}}{2 \cdot 120 \cdot 85})$

Raise $100$ to the power of $2$ .

$A = \arccos (\frac{14400 + 7225 - 1 \cdot 10000}{2 \cdot 120 \cdot 85})$

Multiply $- 1$ by $10000$ .

$A = \arccos (\frac{14400 + 7225 - 10000}{2 \cdot 120 \cdot 85})$

Add $14400$ and $7225$ .

$A = \arccos (\frac{21625 - 10000}{2 \cdot 120 \cdot 85})$

Subtract $10000$ from $21625$ .

$A = \arccos (\frac{11625}{2 \cdot 120 \cdot 85})$

Simplify the denominator.

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Multiply $2$ by $120$ .

$A = \arccos (\frac{11625}{240 \cdot 85})$

Multiply $240$ by $85$ .

$A = \arccos (\frac{11625}{20400})$

Cancel the common factor of $11625$ and $20400$ .

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Factor $75$ out of $11625$ .

$A = \arccos (\frac{75 (155)}{20400})$

Cancel the common factors.

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Factor $75$ out of $20400$ .

$A = \arccos (\frac{75 \cdot 155}{75 \cdot 272})$

Cancel the common factor.

$A = \arccos (\frac{75 \cdot 155}{75 \cdot 272})$

Rewrite the expression.

$A = \arccos (\frac{155}{272})$

Evaluate $\arccos (\frac{155}{272})$ .

$A = 55.26002846$

Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.

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

Solve the equation.

$B = \arccos (\frac{a^{2} + c^{2} - b^{2}}{2 a c})$

Substitute the known values into the equation.

$B = \arccos (\frac{{(100)}^{2} + {(85)}^{2} - {(120)}^{2}}{2 (100) (85)})$

Simplify the results.

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Simplify the numerator.

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Raise $100$ to the power of $2$ .

$B = \arccos (\frac{10000 + {(85)}^{2} - {(120)}^{2}}{2 \cdot 100 \cdot 85})$

Raise $85$ to the power of $2$ .

$B = \arccos (\frac{10000 + 7225 - {(120)}^{2}}{2 \cdot 100 \cdot 85})$

Raise $120$ to the power of $2$ .

$B = \arccos (\frac{10000 + 7225 - 1 \cdot 14400}{2 \cdot 100 \cdot 85})$

Multiply $- 1$ by $14400$ .

$B = \arccos (\frac{10000 + 7225 - 14400}{2 \cdot 100 \cdot 85})$

Add $10000$ and $7225$ .

$B = \arccos (\frac{17225 - 14400}{2 \cdot 100 \cdot 85})$

Subtract $14400$ from $17225$ .

$B = \arccos (\frac{2825}{2 \cdot 100 \cdot 85})$

Simplify the denominator.

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Multiply $2$ by $100$ .

$B = \arccos (\frac{2825}{200 \cdot 85})$

Multiply $200$ by $85$ .

$B = \arccos (\frac{2825}{17000})$

Cancel the common factor of $2825$ and $17000$ .

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Factor $25$ out of $2825$ .

$B = \arccos (\frac{25 (113)}{17000})$

Cancel the common factors.

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Factor $25$ out of $17000$ .

$B = \arccos (\frac{25 \cdot 113}{25 \cdot 680})$

Cancel the common factor.

$B = \arccos (\frac{25 \cdot 113}{25 \cdot 680})$

Rewrite the expression.

$B = \arccos (\frac{113}{680})$

Evaluate $\arccos (\frac{113}{680})$ .

$B = 80.43441514$

The sum of all the angles in a triangle is $180$ degrees.

$55.26002846 + C + 80.43441514 = 180$

Solve the equation for $C$ .

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Add $55.26002846$ and $80.43441514$ .

$C + 135.69444361 = 180$

Move all terms not containing $C$ to the right side of the equation.

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Subtract $135.69444361$ from both sides of the equation.

$C = 180 - 135.69444361$

Subtract $135.69444361$ from $180$ .

$C = 44.30555638$

These are the results for all angles and sides for the given triangle.

$A = 55.26002846$

$B = 80.43441514$

$C = 44.30555638$

$a = 100$

$b = 120$

$c = 85$

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