Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 12 \\ c = & 16 \\ a = & 10 \end{matrix} & \begin{matrix} A = \\ B = \\ C = \end{matrix} \end{matrix}$

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.

$A = \arccos (\frac{{(12)}^{2} + {(16)}^{2} - {(10)}^{2}}{2 (12) (16)})$

Step 4

Simplify the results.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $12$ to the power of $2$ .

$A = \arccos (\frac{144 + 16^{2} - 10^{2}}{2 (12) \cdot 16})$

Step 4.1.2

Raise $16$ to the power of $2$ .

$A = \arccos (\frac{144 + 256 - 10^{2}}{2 (12) \cdot 16})$

Step 4.1.3

Raise $10$ to the power of $2$ .

$A = \arccos (\frac{144 + 256 - 1 \cdot 100}{2 (12) \cdot 16})$

Step 4.1.4

Multiply $- 1$ by $100$ .

$A = \arccos (\frac{144 + 256 - 100}{2 (12) \cdot 16})$

Step 4.1.5

Add $144$ and $256$ .

$A = \arccos (\frac{400 - 100}{2 (12) \cdot 16})$

Step 4.1.6

Subtract $100$ from $400$ .

$A = \arccos (\frac{300}{2 (12) \cdot 16})$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $2$ by $12$ .

$A = \arccos (\frac{300}{24 \cdot 16})$

Step 4.2.2

Multiply $24$ by $16$ .

$A = \arccos (\frac{300}{384})$

Step 4.3

Cancel the common factor of $300$ and $384$ .

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Step 4.3.1

Factor $12$ out of $300$ .

$A = \arccos (\frac{12 (25)}{384})$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $12$ out of $384$ .

$A = \arccos (\frac{12 \cdot 25}{12 \cdot 32})$

Step 4.3.2.2

Cancel the common factor.

$A = \arccos (\frac{12 \cdot 25}{12 \cdot 32})$

Step 4.3.2.3

Rewrite the expression.

$A = \arccos (\frac{25}{32})$

Step 4.4

Evaluate $\arccos (\frac{25}{32})$ .

$A = 38.62483287$

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.

$B = \arccos (\frac{{(10)}^{2} + {(16)}^{2} - {(12)}^{2}}{2 (10) (16)})$

Step 8

Simplify the results.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $10$ to the power of $2$ .

$B = \arccos (\frac{100 + 16^{2} - 12^{2}}{2 (10) \cdot 16})$

Step 8.1.2

Raise $16$ to the power of $2$ .

$B = \arccos (\frac{100 + 256 - 12^{2}}{2 (10) \cdot 16})$

Step 8.1.3

Raise $12$ to the power of $2$ .

$B = \arccos (\frac{100 + 256 - 1 \cdot 144}{2 (10) \cdot 16})$

Step 8.1.4

Multiply $- 1$ by $144$ .

$B = \arccos (\frac{100 + 256 - 144}{2 (10) \cdot 16})$

Step 8.1.5

Add $100$ and $256$ .

$B = \arccos (\frac{356 - 144}{2 (10) \cdot 16})$

Step 8.1.6

Subtract $144$ from $356$ .

$B = \arccos (\frac{212}{2 (10) \cdot 16})$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $2$ by $10$ .

$B = \arccos (\frac{212}{20 \cdot 16})$

Step 8.2.2

Multiply $20$ by $16$ .

$B = \arccos (\frac{212}{320})$

Step 8.3

Cancel the common factor of $212$ and $320$ .

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Step 8.3.1

Factor $4$ out of $212$ .

$B = \arccos (\frac{4 (53)}{320})$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor $4$ out of $320$ .

$B = \arccos (\frac{4 \cdot 53}{4 \cdot 80})$

Step 8.3.2.2

Cancel the common factor.

$B = \arccos (\frac{4 \cdot 53}{4 \cdot 80})$

Step 8.3.2.3

Rewrite the expression.

$B = \arccos (\frac{53}{80})$

Step 8.4

Evaluate $\arccos (\frac{53}{80})$ .

$B = 48.50918314$

Step 9

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

$38.62483287 + C + 48.50918314 = 180$

Step 10

Solve the equation for $C$ .

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Step 10.1

Add $38.62483287$ and $48.50918314$ .

$C + 87.13401601 = 180$

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 $87.13401601$ from both sides of the equation.

$C = 180 - 87.13401601$

Step 10.2.2

Subtract $87.13401601$ from $180$ .

$C = 92.86598398$

Step 11

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

$A = 38.62483287$

$B = 48.50918314$

$C = 92.86598398$

$a = 10$

$b = 12$

$c = 16$