Precalculus Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 8 \\ c = & 17 \\ a = & 15 \end{matrix} & \begin{matrix} A = \\ B = \\ C = \end{matrix} \end{matrix}$

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.

$A = \arccos (\frac{{(8)}^{2} + {(17)}^{2} - {(15)}^{2}}{2 (8) (17)})$

Step 4

Simplify the results.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $8$ to the power of $2$ .

$A = \arccos (\frac{64 + 17^{2} - 15^{2}}{2 (8) \cdot 17})$

Step 4.1.2

Raise $17$ to the power of $2$ .

$A = \arccos (\frac{64 + 289 - 15^{2}}{2 (8) \cdot 17})$

Step 4.1.3

Raise $15$ to the power of $2$ .

$A = \arccos (\frac{64 + 289 - 1 \cdot 225}{2 (8) \cdot 17})$

Step 4.1.4

Multiply $- 1$ by $225$ .

$A = \arccos (\frac{64 + 289 - 225}{2 (8) \cdot 17})$

Step 4.1.5

Add $64$ and $289$ .

$A = \arccos (\frac{353 - 225}{2 (8) \cdot 17})$

Step 4.1.6

Subtract $225$ from $353$ .

$A = \arccos (\frac{128}{2 (8) \cdot 17})$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $2$ by $8$ .

$A = \arccos (\frac{128}{16 \cdot 17})$

Step 4.2.2

Multiply $16$ by $17$ .

$A = \arccos (\frac{128}{272})$

Step 4.3

Cancel the common factor of $128$ and $272$ .

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Step 4.3.1

Factor $16$ out of $128$ .

$A = \arccos (\frac{16 (8)}{272})$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $16$ out of $272$ .

$A = \arccos (\frac{16 \cdot 8}{16 \cdot 17})$

Step 4.3.2.2

Cancel the common factor.

$A = \arccos (\frac{16 \cdot 8}{16 \cdot 17})$

Step 4.3.2.3

Rewrite the expression.

$A = \arccos (\frac{8}{17})$

Step 4.4

Evaluate $\arccos (\frac{8}{17})$ .

$A = 61.92751306$

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.

$B = \arccos (\frac{{(15)}^{2} + {(17)}^{2} - {(8)}^{2}}{2 (15) (17)})$

Step 8

Simplify the results.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $15$ to the power of $2$ .

$B = \arccos (\frac{225 + 17^{2} - 8^{2}}{2 (15) \cdot 17})$

Step 8.1.2

Raise $17$ to the power of $2$ .

$B = \arccos (\frac{225 + 289 - 8^{2}}{2 (15) \cdot 17})$

Step 8.1.3

Raise $8$ to the power of $2$ .

$B = \arccos (\frac{225 + 289 - 1 \cdot 64}{2 (15) \cdot 17})$

Step 8.1.4

Multiply $- 1$ by $64$ .

$B = \arccos (\frac{225 + 289 - 64}{2 (15) \cdot 17})$

Step 8.1.5

Add $225$ and $289$ .

$B = \arccos (\frac{514 - 64}{2 (15) \cdot 17})$

Step 8.1.6

Subtract $64$ from $514$ .

$B = \arccos (\frac{450}{2 (15) \cdot 17})$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $2$ by $15$ .

$B = \arccos (\frac{450}{30 \cdot 17})$

Step 8.2.2

Multiply $30$ by $17$ .

$B = \arccos (\frac{450}{510})$

Step 8.3

Cancel the common factor of $450$ and $510$ .

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Step 8.3.1

Factor $30$ out of $450$ .

$B = \arccos (\frac{30 (15)}{510})$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor $30$ out of $510$ .

$B = \arccos (\frac{30 \cdot 15}{30 \cdot 17})$

Step 8.3.2.2

Cancel the common factor.

$B = \arccos (\frac{30 \cdot 15}{30 \cdot 17})$

Step 8.3.2.3

Rewrite the expression.

$B = \arccos (\frac{15}{17})$

Step 8.4

Evaluate $\arccos (\frac{15}{17})$ .

$B = 28.07248693$

Step 9

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

$61.92751306 + C + 28.07248693 = 180$

Step 10

Solve the equation for $C$ .

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Step 10.1

Add $61.92751306$ and $28.07248693$ .

$C + 90 = 180$

Step 10.2

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

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Step 10.2.1

Subtract $90$ from both sides of the equation.

$C = 180 - 90$

Step 10.2.2

Subtract $90$ from $180$ .

$C = 90$

Step 11

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

$A = 61.92751306$

$B = 28.07248693$

$C = 90$

$a = 15$

$b = 8$

$c = 17$