Trigonometry Examples

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\begin{matrix} Side & Angle \begin{matrix} b = & 10 c = & 26 a = & 24 \end{matrix} & \begin{matrix} A = B = C = \end{matrix} \end{matrix}

$\begin{matrix} Side & Angle \\ \begin{matrix} b = & 10 \\ c = & 26 \\ a = & 24 \end{matrix} & \begin{matrix} A = \\ B = \\ C = \end{matrix} \end{matrix}$

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{{(10)}^{2} + {(26)}^{2} - {(24)}^{2}}{2 (10) (26)})

$A = \arccos (\frac{{(10)}^{2} + {(26)}^{2} - {(24)}^{2}}{2 (10) (26)})$

Step 4

Simplify the results.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise

10

$10$ to the power of

2

$2$ .

A = arccos (\frac{100 + 26^{2} - 24^{2}}{2 (10) \cdot 26})

$A = \arccos (\frac{100 + 26^{2} - 24^{2}}{2 (10) \cdot 26})$

Step 4.1.2

Raise

26

$26$ to the power of

2

$2$ .

A = arccos (\frac{100 + 676 - 24^{2}}{2 (10) \cdot 26})

$A = \arccos (\frac{100 + 676 - 24^{2}}{2 (10) \cdot 26})$

Step 4.1.3

Raise

24

$24$ to the power of

2

$2$ .

A = arccos (\frac{100 + 676 - 1 \cdot 576}{2 (10) \cdot 26})

$A = \arccos (\frac{100 + 676 - 1 \cdot 576}{2 (10) \cdot 26})$

Step 4.1.4

Multiply

- 1

$- 1$ by

576

$576$ .

A = arccos (\frac{100 + 676 - 576}{2 (10) \cdot 26})

$A = \arccos (\frac{100 + 676 - 576}{2 (10) \cdot 26})$

Step 4.1.5

Add

100

$100$ and

676

$676$ .

A = arccos (\frac{776 - 576}{2 (10) \cdot 26})

$A = \arccos (\frac{776 - 576}{2 (10) \cdot 26})$

Step 4.1.6

Subtract

576

$576$ from

776

$776$ .

A = arccos (\frac{200}{2 (10) \cdot 26})

$A = \arccos (\frac{200}{2 (10) \cdot 26})$

A = arccos (\frac{200}{2 (10) \cdot 26})

$A = \arccos (\frac{200}{2 (10) \cdot 26})$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply

2

$2$ by

10

$10$ .

A = arccos (\frac{200}{20 \cdot 26})

$A = \arccos (\frac{200}{20 \cdot 26})$

Step 4.2.2

Multiply

20

$20$ by

26

$26$ .

A = arccos (\frac{200}{520})

$A = \arccos (\frac{200}{520})$

A = arccos (\frac{200}{520})

$A = \arccos (\frac{200}{520})$

Step 4.3

Cancel the common factor of

200

$200$ and

520

$520$ .

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Step 4.3.1

Factor

40

$40$ out of

200

$200$ .

A = arccos (\frac{40 (5)}{520})

$A = \arccos (\frac{40 (5)}{520})$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor

40

$40$ out of

520

$520$ .

A = arccos (\frac{40 \cdot 5}{40 \cdot 13})

$A = \arccos (\frac{40 \cdot 5}{40 \cdot 13})$

Step 4.3.2.2

Cancel the common factor.

$A = \arccos (\frac{40 \cdot 5}{40 \cdot 13})$

Step 4.3.2.3

Rewrite the expression.

$A = \arccos (\frac{5}{13})$

Step 4.4

Evaluate

$\arccos (\frac{5}{13})$ .

$A = 67.38013505$

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.

$B = \arccos (\frac{{(24)}^{2} + {(26)}^{2} - {(10)}^{2}}{2 (24) (26)})$

Step 8

Simplify the results.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise

$24$ to the power of

$2$ .

$B = \arccos (\frac{576 + 26^{2} - 10^{2}}{2 (24) \cdot 26})$

Step 8.1.2

Raise

$26$ to the power of

$2$ .

$B = \arccos (\frac{576 + 676 - 10^{2}}{2 (24) \cdot 26})$

Step 8.1.3

Raise

$10$ to the power of

$2$ .

$B = \arccos (\frac{576 + 676 - 1 \cdot 100}{2 (24) \cdot 26})$

Step 8.1.4

Multiply

$- 1$ by

$100$ .

$B = \arccos (\frac{576 + 676 - 100}{2 (24) \cdot 26})$

Step 8.1.5

Add

$576$ and

$676$ .

$B = \arccos (\frac{1252 - 100}{2 (24) \cdot 26})$

Step 8.1.6

Subtract

$100$ from

$1252$ .

$B = \arccos (\frac{1152}{2 (24) \cdot 26})$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply

$2$ by

$24$ .

$B = \arccos (\frac{1152}{48 \cdot 26})$

Step 8.2.2

Multiply

$48$ by

$26$ .

$B = \arccos (\frac{1152}{1248})$

Step 8.3

Cancel the common factor of

$1152$ and

$1248$ .

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Step 8.3.1

Factor

$96$ out of

$1152$ .

$B = \arccos (\frac{96 (12)}{1248})$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor

$96$ out of

$1248$ .

$B = \arccos (\frac{96 \cdot 12}{96 \cdot 13})$

Step 8.3.2.2

Cancel the common factor.

$B = \arccos (\frac{96 \cdot 12}{96 \cdot 13})$

Step 8.3.2.3

Rewrite the expression.

$B = \arccos (\frac{12}{13})$

Step 8.4

Evaluate

$\arccos (\frac{12}{13})$ .

$B = 22.61986494$

Step 9

The sum of all the angles in a triangle is

$180$ degrees.

$67.38013505 + C + 22.61986494 = 180$

Step 10

Solve the equation for

$C$ .

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Step 10.1

Add

$67.38013505$ and

$22.61986494$ .

$C + 90 = 180$

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.

$A = 67.38013505$

$B = 22.61986494$

$C = 90$

$a = 24$

$b = 10$

$c = 26$