Solve the Triangle a=20 , b=21 , c=29

$a = 20$ , $b = 21$ , $c = 29$

Step 1

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

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

Step 2

Solve the equation.

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

Step 3

Substitute the known values into the equation.

$A = \arccos (\frac{{(21)}^{2} + {(29)}^{2} - {(20)}^{2}}{2 (21) (29)})$

Step 4

Simplify the results.

Simplify the numerator.

Raise $21$ to the power of $2$ .

$A = \arccos (\frac{441 + 29^{2} - 20^{2}}{2 (21) \cdot 29})$

Raise $29$ to the power of $2$ .

$A = \arccos (\frac{441 + 841 - 20^{2}}{2 (21) \cdot 29})$

Raise $20$ to the power of $2$ .

$A = \arccos (\frac{441 + 841 - 1 \cdot 400}{2 (21) \cdot 29})$

Multiply $- 1$ by $400$ .

$A = \arccos (\frac{441 + 841 - 400}{2 (21) \cdot 29})$

Add $441$ and $841$ .

$A = \arccos (\frac{1282 - 400}{2 (21) \cdot 29})$

Subtract $400$ from $1282$ .

$A = \arccos (\frac{882}{2 (21) \cdot 29})$

Simplify the denominator.

Multiply $2$ by $21$ .

$A = \arccos (\frac{882}{42 \cdot 29})$

Multiply $42$ by $29$ .

$A = \arccos (\frac{882}{1218})$

Cancel the common factor of $882$ and $1218$ .

Factor $42$ out of $882$ .

$A = \arccos (\frac{42 (21)}{1218})$

Cancel the common factors.

Factor $42$ out of $1218$ .

$A = \arccos (\frac{42 \cdot 21}{42 \cdot 29})$

Cancel the common factor.

$A = \arccos (\frac{42 \cdot 21}{42 \cdot 29})$

Rewrite the expression.

$A = \arccos (\frac{21}{29})$

Evaluate $\arccos (\frac{21}{29})$ .

$A = 43.60281897$

Step 5

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

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

Step 6

Solve the equation.

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

Step 7

Substitute the known values into the equation.

$B = \arccos (\frac{{(20)}^{2} + {(29)}^{2} - {(21)}^{2}}{2 (20) (29)})$

Step 8

Simplify the results.

Simplify the numerator.

Raise $20$ to the power of $2$ .

$B = \arccos (\frac{400 + 29^{2} - 21^{2}}{2 (20) \cdot 29})$

Raise $29$ to the power of $2$ .

$B = \arccos (\frac{400 + 841 - 21^{2}}{2 (20) \cdot 29})$

Raise $21$ to the power of $2$ .

$B = \arccos (\frac{400 + 841 - 1 \cdot 441}{2 (20) \cdot 29})$

Multiply $- 1$ by $441$ .

$B = \arccos (\frac{400 + 841 - 441}{2 (20) \cdot 29})$

Add $400$ and $841$ .

$B = \arccos (\frac{1241 - 441}{2 (20) \cdot 29})$

Subtract $441$ from $1241$ .

$B = \arccos (\frac{800}{2 (20) \cdot 29})$

Simplify the denominator.

Multiply $2$ by $20$ .

$B = \arccos (\frac{800}{40 \cdot 29})$

Multiply $40$ by $29$ .

$B = \arccos (\frac{800}{1160})$

Cancel the common factor of $800$ and $1160$ .

Factor $40$ out of $800$ .

$B = \arccos (\frac{40 (20)}{1160})$

Cancel the common factors.

Factor $40$ out of $1160$ .

$B = \arccos (\frac{40 \cdot 20}{40 \cdot 29})$

Cancel the common factor.

$B = \arccos (\frac{40 \cdot 20}{40 \cdot 29})$

Rewrite the expression.

$B = \arccos (\frac{20}{29})$

Evaluate $\arccos (\frac{20}{29})$ .

$B = 46.39718102$

Step 9

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

$43.60281897 + C + 46.39718102 = 180$

Step 10

Solve the equation for $C$ .

Add $43.60281897$ and $46.39718102$ .

$C + 90 = 180$

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

Subtract $90$ from both sides of the equation.

$C = 180 - 90$

Subtract $90$ from $180$ .

$C = 90$

Step 11

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

$A = 43.60281897$

$B = 46.39718102$

$C = 90$

$a = 20$

$b = 21$

$c = 29$

Solve the Triangle a=20 , b=21 , c=29

Related Questions

Support LCMGCF.com by disabling your adblocker.