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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Name

Name	one billion eight hundred twelve million one hundred forty-six thousand two hundred thirty-one

Interesting facts

1812146231 has 4 divisors, whose sum is 2071024272
The reverse of 1812146231 is 1326412181
Previous prime number is 7

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 29
Digital Root 2

Name

Name	two hundred six million five hundred three thousand eighteen

Interesting facts

206503018 has 4 divisors, whose sum is 309754530
The reverse of 206503018 is 810305602
Previous prime number is 2

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 25
Digital Root 7

Name

Name	two billion fifty-seven million three hundred sixty-one thousand three hundred ninety-four

Interesting facts

2057361394 has 32 divisors, whose sum is 3488334336
The reverse of 2057361394 is 4931637502
Previous prime number is 113

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 40
Digital Root 4

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