Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 9 \\ c = & 6 \\ a = & 14 \end{matrix} & \begin{matrix} A = \\ B = \\ C = \end{matrix} \end{matrix}$

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.

$A = \arccos (\frac{{(9)}^{2} + {(6)}^{2} - {(14)}^{2}}{2 (9) (6)})$

Step 4

Simplify the results.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $9$ to the power of $2$ .

$A = \arccos (\frac{81 + 6^{2} - 14^{2}}{2 (9) \cdot 6})$

Step 4.1.2

Raise $6$ to the power of $2$ .

$A = \arccos (\frac{81 + 36 - 14^{2}}{2 (9) \cdot 6})$

Step 4.1.3

Raise $14$ to the power of $2$ .

$A = \arccos (\frac{81 + 36 - 1 \cdot 196}{2 (9) \cdot 6})$

Step 4.1.4

Multiply $- 1$ by $196$ .

$A = \arccos (\frac{81 + 36 - 196}{2 (9) \cdot 6})$

Step 4.1.5

Add $81$ and $36$ .

$A = \arccos (\frac{117 - 196}{2 (9) \cdot 6})$

Step 4.1.6

Subtract $196$ from $117$ .

$A = \arccos (\frac{- 79}{2 (9) \cdot 6})$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $2$ by $9$ .

$A = \arccos (\frac{- 79}{18 \cdot 6})$

Step 4.2.2

Multiply $18$ by $6$ .

$A = \arccos (\frac{- 79}{108})$

Step 4.3

Move the negative in front of the fraction.

$A = \arccos (- \frac{79}{108})$

Step 4.4

Evaluate $\arccos (- \frac{79}{108})$ .

$A = 137.01073607$

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.

$B = \arccos (\frac{{(14)}^{2} + {(6)}^{2} - {(9)}^{2}}{2 (14) (6)})$

Step 8

Simplify the results.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $14$ to the power of $2$ .

$B = \arccos (\frac{196 + 6^{2} - 9^{2}}{2 (14) \cdot 6})$

Step 8.1.2

Raise $6$ to the power of $2$ .

$B = \arccos (\frac{196 + 36 - 9^{2}}{2 (14) \cdot 6})$

Step 8.1.3

Raise $9$ to the power of $2$ .

$B = \arccos (\frac{196 + 36 - 1 \cdot 81}{2 (14) \cdot 6})$

Step 8.1.4

Multiply $- 1$ by $81$ .

$B = \arccos (\frac{196 + 36 - 81}{2 (14) \cdot 6})$

Step 8.1.5

Add $196$ and $36$ .

$B = \arccos (\frac{232 - 81}{2 (14) \cdot 6})$

Step 8.1.6

Subtract $81$ from $232$ .

$B = \arccos (\frac{151}{2 (14) \cdot 6})$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $2$ by $14$ .

$B = \arccos (\frac{151}{28 \cdot 6})$

Step 8.2.2

Multiply $28$ by $6$ .

$B = \arccos (\frac{151}{168})$

Step 8.3

Evaluate $\arccos (\frac{151}{168})$ .

$B = 25.99797699$

Step 9

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

$137.01073607 + C + 25.99797699 = 180$

Step 10

Solve the equation for $C$ .

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Step 10.1

Add $137.01073607$ and $25.99797699$ .

$C + 163.00871306 = 180$

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 $163.00871306$ from both sides of the equation.

$C = 180 - 163.00871306$

Step 10.2.2

Subtract $163.00871306$ from $180$ .

$C = 16.99128693$

Step 11

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

$A = 137.01073607$

$B = 25.99797699$

$C = 16.99128693$

$a = 14$

$b = 9$

$c = 6$