Solve the Triangle a=60 , b=67 , c=72

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Trigonometry

$a = 60$ , $b = 67$ , $c = 72$

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.

$A = \arccos (\frac{{(67)}^{2} + {(72)}^{2} - {(60)}^{2}}{2 (67) (72)})$

Simplify the results.

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Simplify the numerator.

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Raise $67$ to the power of $2$ .

$A = \arccos (\frac{4489 + {(72)}^{2} - {(60)}^{2}}{2 \cdot 67 \cdot 72})$

Raise $72$ to the power of $2$ .

$A = \arccos (\frac{4489 + 5184 - {(60)}^{2}}{2 \cdot 67 \cdot 72})$

Raise $60$ to the power of $2$ .

$A = \arccos (\frac{4489 + 5184 - 1 \cdot 3600}{2 \cdot 67 \cdot 72})$

Multiply $- 1$ by $3600$ .

$A = \arccos (\frac{4489 + 5184 - 3600}{2 \cdot 67 \cdot 72})$

Add $4489$ and $5184$ .

$A = \arccos (\frac{9673 - 3600}{2 \cdot 67 \cdot 72})$

Subtract $3600$ from $9673$ .

$A = \arccos (\frac{6073}{2 \cdot 67 \cdot 72})$

Simplify the denominator.

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Multiply $2$ by $67$ .

$A = \arccos (\frac{6073}{134 \cdot 72})$

Multiply $134$ by $72$ .

$A = \arccos (\frac{6073}{9648})$

Evaluate $\arccos (\frac{6073}{9648})$ .

$A = 50.98993634$

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.

$B = \arccos (\frac{{(60)}^{2} + {(72)}^{2} - {(67)}^{2}}{2 (60) (72)})$

Simplify the results.

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Simplify the numerator.

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Raise $60$ to the power of $2$ .

$B = \arccos (\frac{3600 + {(72)}^{2} - {(67)}^{2}}{2 \cdot 60 \cdot 72})$

Raise $72$ to the power of $2$ .

$B = \arccos (\frac{3600 + 5184 - {(67)}^{2}}{2 \cdot 60 \cdot 72})$

Raise $67$ to the power of $2$ .

$B = \arccos (\frac{3600 + 5184 - 1 \cdot 4489}{2 \cdot 60 \cdot 72})$

Multiply $- 1$ by $4489$ .

$B = \arccos (\frac{3600 + 5184 - 4489}{2 \cdot 60 \cdot 72})$

Add $3600$ and $5184$ .

$B = \arccos (\frac{8784 - 4489}{2 \cdot 60 \cdot 72})$

Subtract $4489$ from $8784$ .

$B = \arccos (\frac{4295}{2 \cdot 60 \cdot 72})$

Simplify the denominator.

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Multiply $2$ by $60$ .

$B = \arccos (\frac{4295}{120 \cdot 72})$

Multiply $120$ by $72$ .

$B = \arccos (\frac{4295}{8640})$

Cancel the common factor of $4295$ and $8640$ .

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Factor $5$ out of $4295$ .

$B = \arccos (\frac{5 (859)}{8640})$

Cancel the common factors.

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Factor $5$ out of $8640$ .

$B = \arccos (\frac{5 \cdot 859}{5 \cdot 1728})$

Cancel the common factor.

$B = \arccos (\frac{5 \cdot 859}{5 \cdot 1728})$

Rewrite the expression.

$B = \arccos (\frac{859}{1728})$

Evaluate $\arccos (\frac{859}{1728})$ .

$B = 60.19124971$

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

$50.98993634 + C + 60.19124971 = 180$

Solve the equation for $C$ .

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Add $50.98993634$ and $60.19124971$ .

$C + 111.18118606 = 180$

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

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Subtract $111.18118606$ from both sides of the equation.

$C = 180 - 111.18118606$

Subtract $111.18118606$ from $180$ .

$C = 68.81881393$

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

$A = 50.98993634$

$B = 60.19124971$

$C = 68.81881393$

$a = 60$

$b = 67$

$c = 72$

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