Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 6 \\ c = & 10 \\ a = & 8 \end{matrix} & \begin{matrix} A = \\ B = \\ C = \end{matrix} \end{matrix}$

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.

$A = \arccos (\frac{{(6)}^{2} + {(10)}^{2} - {(8)}^{2}}{2 (6) (10)})$

Step 4

Simplify the results.

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Simplify the numerator.

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

$A = \arccos (\frac{36 + 10^{2} - 8^{2}}{2 (6) \cdot 10})$

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

$A = \arccos (\frac{36 + 100 - 8^{2}}{2 (6) \cdot 10})$

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

$A = \arccos (\frac{36 + 100 - 1 \cdot 64}{2 (6) \cdot 10})$

Multiply $- 1$ by $64$ .

$A = \arccos (\frac{36 + 100 - 64}{2 (6) \cdot 10})$

Add $36$ and $100$ .

$A = \arccos (\frac{136 - 64}{2 (6) \cdot 10})$

Subtract $64$ from $136$ .

$A = \arccos (\frac{72}{2 (6) \cdot 10})$

Simplify the denominator.

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Multiply $2$ by $6$ .

$A = \arccos (\frac{72}{12 \cdot 10})$

Multiply $12$ by $10$ .

$A = \arccos (\frac{72}{120})$

Cancel the common factor of $72$ and $120$ .

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Factor $24$ out of $72$ .

$A = \arccos (\frac{24 (3)}{120})$

Cancel the common factors.

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Factor $24$ out of $120$ .

$A = \arccos (\frac{24 \cdot 3}{24 \cdot 5})$

Cancel the common factor.

$A = \arccos (\frac{24 \cdot 3}{24 \cdot 5})$

Rewrite the expression.

$A = \arccos (\frac{3}{5})$

Evaluate $\arccos (\frac{3}{5})$ .

$A = 53.13010235$

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.

$B = \arccos (\frac{{(8)}^{2} + {(10)}^{2} - {(6)}^{2}}{2 (8) (10)})$

Step 8

Simplify the results.

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Simplify the numerator.

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

$B = \arccos (\frac{64 + 10^{2} - 6^{2}}{2 (8) \cdot 10})$

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

$B = \arccos (\frac{64 + 100 - 6^{2}}{2 (8) \cdot 10})$

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

$B = \arccos (\frac{64 + 100 - 1 \cdot 36}{2 (8) \cdot 10})$

Multiply $- 1$ by $36$ .

$B = \arccos (\frac{64 + 100 - 36}{2 (8) \cdot 10})$

Add $64$ and $100$ .

$B = \arccos (\frac{164 - 36}{2 (8) \cdot 10})$

Subtract $36$ from $164$ .

$B = \arccos (\frac{128}{2 (8) \cdot 10})$

Simplify the denominator.

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Multiply $2$ by $8$ .

$B = \arccos (\frac{128}{16 \cdot 10})$

Multiply $16$ by $10$ .

$B = \arccos (\frac{128}{160})$

Cancel the common factor of $128$ and $160$ .

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Factor $32$ out of $128$ .

$B = \arccos (\frac{32 (4)}{160})$

Cancel the common factors.

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Factor $32$ out of $160$ .

$B = \arccos (\frac{32 \cdot 4}{32 \cdot 5})$

Cancel the common factor.

$B = \arccos (\frac{32 \cdot 4}{32 \cdot 5})$

Rewrite the expression.

$B = \arccos (\frac{4}{5})$

Evaluate $\arccos (\frac{4}{5})$ .

$B = 36.86989764$

Step 9

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

$53.13010235 + C + 36.86989764 = 180$

Step 10

Solve the equation for $C$ .

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Add $53.13010235$ and $36.86989764$ .

$C + 90 = 180$

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

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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.

$A = 53.13010235$

$B = 36.86989764$

$C = 90$

$a = 8$

$b = 6$

$c = 10$