Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 13.7 \\ c = & 12.2 \\ a = & 11.4 \end{matrix} & \begin{matrix} A = \\ B = \\ C = \end{matrix} \end{matrix}$

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.

$A = \arccos (\frac{{(13.7)}^{2} + {(12.2)}^{2} - {(11.4)}^{2}}{2 (13.7) (12.2)})$

Step 4

Simplify the results.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $13.7$ to the power of $2$ .

$A = \arccos (\frac{187.69 + {12.2}^{2} - {11.4}^{2}}{2 (13.7) \cdot 12.2})$

Step 4.1.2

Raise $12.2$ to the power of $2$ .

$A = \arccos (\frac{187.69 + 148.84 - {11.4}^{2}}{2 (13.7) \cdot 12.2})$

Step 4.1.3

Raise $11.4$ to the power of $2$ .

$A = \arccos (\frac{187.69 + 148.84 - 1 \cdot 129.96}{2 (13.7) \cdot 12.2})$

Step 4.1.4

Multiply $- 1$ by $129.96$ .

$A = \arccos (\frac{187.69 + 148.84 - 129.96}{2 (13.7) \cdot 12.2})$

Step 4.1.5

Add $187.69$ and $148.84$ .

$A = \arccos (\frac{336.53 - 129.96}{2 (13.7) \cdot 12.2})$

Step 4.1.6

Subtract $129.96$ from $336.53$ .

$A = \arccos (\frac{206.57}{2 (13.7) \cdot 12.2})$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $2$ by $13.7$ .

$A = \arccos (\frac{206.57}{27.4 \cdot 12.2})$

Step 4.2.2

Multiply $27.4$ by $12.2$ .

$A = \arccos (\frac{206.57}{334.28})$

Step 4.3

Divide $206.57$ by $334.28$ .

$A = \arccos (0.617955)$

Step 4.4

Evaluate $\arccos (0.617955)$ .

$A = 51.83304838$

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.

$B = \arccos (\frac{{(11.4)}^{2} + {(12.2)}^{2} - {(13.7)}^{2}}{2 (11.4) (12.2)})$

Step 8

Simplify the results.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $11.4$ to the power of $2$ .

$B = \arccos (\frac{129.96 + {12.2}^{2} - {13.7}^{2}}{2 (11.4) \cdot 12.2})$

Step 8.1.2

Raise $12.2$ to the power of $2$ .

$B = \arccos (\frac{129.96 + 148.84 - {13.7}^{2}}{2 (11.4) \cdot 12.2})$

Step 8.1.3

Raise $13.7$ to the power of $2$ .

$B = \arccos (\frac{129.96 + 148.84 - 1 \cdot 187.69}{2 (11.4) \cdot 12.2})$

Step 8.1.4

Multiply $- 1$ by $187.69$ .

$B = \arccos (\frac{129.96 + 148.84 - 187.69}{2 (11.4) \cdot 12.2})$

Step 8.1.5

Add $129.96$ and $148.84$ .

$B = \arccos (\frac{278.8 - 187.69}{2 (11.4) \cdot 12.2})$

Step 8.1.6

Subtract $187.69$ from $278.8$ .

$B = \arccos (\frac{91.11}{2 (11.4) \cdot 12.2})$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $2$ by $11.4$ .

$B = \arccos (\frac{91.11}{22.8 \cdot 12.2})$

Step 8.2.2

Multiply $22.8$ by $12.2$ .

$B = \arccos (\frac{91.11}{278.16})$

Step 8.3

Divide $91.11$ by $278.16$ .

$B = \arccos (0.32754529)$

Step 8.4

Evaluate $\arccos (0.32754529)$ .

$B = 70.8801474$

Step 9

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

$51.83304838 + C + 70.8801474 = 180$

Step 10

Solve the equation for $C$ .

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Step 10.1

Add $51.83304838$ and $70.8801474$ .

$C + 122.71319579 = 180$

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 $122.71319579$ from both sides of the equation.

$C = 180 - 122.71319579$

Step 10.2.2

Subtract $122.71319579$ from $180$ .

$C = 57.2868042$

Step 11

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

$A = 51.83304838$

$B = 70.8801474$

$C = 57.2868042$

$a = 11.4$

$b = 13.7$

$c = 12.2$