Trigonometry Examples

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

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

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)

$a^{2} = b^{2} + c^{2} - 2 b c \cos (A)$

Step 2

Solve the equation.

A = arccos (\frac{b^{2} + c^{2} - a^{2}}{2 b c})

$A = \arccos (\frac{b^{2} + c^{2} - a^{2}}{2 b c})$

Step 3

Substitute the known values into the equation.

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

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

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

$8$ to the power of

2

$2$ .

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

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

Step 4.1.2

Raise

10

$10$ to the power of

2

$2$ .

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

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

Step 4.1.3

Raise

6

$6$ to the power of

2

$2$ .

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

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

Step 4.1.4

Multiply

- 1

$- 1$ by

36

$36$ .

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

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

Step 4.1.5

Add

64

$64$ and

100

$100$ .

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

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

Step 4.1.6

Subtract

36

$36$ from

164

$164$ .

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

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

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

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

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply

2

$2$ by

8

$8$ .

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

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

Step 4.2.2

Multiply

16

$16$ by

10

$10$ .

A = arccos (\frac{128}{160})

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

A = arccos (\frac{128}{160})

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

Step 4.3

Cancel the common factor of

128

$128$ and

160

$160$ .

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Step 4.3.1

Factor

32

$32$ out of

128

$128$ .

A = arccos (\frac{32 (4)}{160})

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

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor

32

$32$ out of

160

$160$ .

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

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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

Step 4.4

Evaluate

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

$A = 36.86989764$

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.

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

Step 8

Simplify the results.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise

$6$ to the power of

$2$ .

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

Step 8.1.2

Raise

$10$ to the power of

$2$ .

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

Step 8.1.3

Raise

$8$ to the power of

$2$ .

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

Step 8.1.4

Multiply

$- 1$ by

$64$ .

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

Step 8.1.5

Add

$36$ and

$100$ .

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

Step 8.1.6

Subtract

$64$ from

$136$ .

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

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply

$2$ by

$6$ .

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

Step 8.2.2

Multiply

$12$ by

$10$ .

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

Step 8.3

Cancel the common factor of

$72$ and

$120$ .

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Step 8.3.1

Factor

$24$ out of

$72$ .

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

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor

$24$ out of

$120$ .

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

Step 8.3.2.2

Cancel the common factor.

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

Step 8.3.2.3

Rewrite the expression.

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

Step 8.4

Evaluate

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

$B = 53.13010235$

Step 9

The sum of all the angles in a triangle is

$180$ degrees.

$36.86989764 + C + 53.13010235 = 180$

Step 10

Solve the equation for

$C$ .

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Step 10.1

Add

$36.86989764$ and

$53.13010235$ .

$C + 90 = 180$

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.

$A = 36.86989764$

$B = 53.13010235$

$C = 90$

$a = 6$

$b = 8$

$c = 10$