Trigonometry Examples

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\begin{matrix} Side & Angle \begin{matrix} b = & 35 c = & 37 a = & 12 \end{matrix} & \begin{matrix} A = B = C = \end{matrix} \end{matrix}

$\begin{matrix} Side & Angle \\ \begin{matrix} b = & 35 \\ c = & 37 \\ a = & 12 \end{matrix} & \begin{matrix} A = \\ B = \\ C = \end{matrix} \end{matrix}$

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{{(35)}^{2} + {(37)}^{2} - {(12)}^{2}}{2 (35) (37)})

$A = \arccos (\frac{{(35)}^{2} + {(37)}^{2} - {(12)}^{2}}{2 (35) (37)})$

Step 4

Simplify the results.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise

35

$35$ to the power of

2

$2$ .

A = arccos (\frac{1225 + 37^{2} - 12^{2}}{2 (35) \cdot 37})

$A = \arccos (\frac{1225 + 37^{2} - 12^{2}}{2 (35) \cdot 37})$

Step 4.1.2

Raise

37

$37$ to the power of

2

$2$ .

A = arccos (\frac{1225 + 1369 - 12^{2}}{2 (35) \cdot 37})

$A = \arccos (\frac{1225 + 1369 - 12^{2}}{2 (35) \cdot 37})$

Step 4.1.3

Raise

12

$12$ to the power of

2

$2$ .

A = arccos (\frac{1225 + 1369 - 1 \cdot 144}{2 (35) \cdot 37})

$A = \arccos (\frac{1225 + 1369 - 1 \cdot 144}{2 (35) \cdot 37})$

Step 4.1.4

Multiply

- 1

$- 1$ by

144

$144$ .

A = arccos (\frac{1225 + 1369 - 144}{2 (35) \cdot 37})

$A = \arccos (\frac{1225 + 1369 - 144}{2 (35) \cdot 37})$

Step 4.1.5

Add

1225

$1225$ and

1369

$1369$ .

A = arccos (\frac{2594 - 144}{2 (35) \cdot 37})

$A = \arccos (\frac{2594 - 144}{2 (35) \cdot 37})$

Step 4.1.6

Subtract

144

$144$ from

2594

$2594$ .

A = arccos (\frac{2450}{2 (35) \cdot 37})

$A = \arccos (\frac{2450}{2 (35) \cdot 37})$

A = arccos (\frac{2450}{2 (35) \cdot 37})

$A = \arccos (\frac{2450}{2 (35) \cdot 37})$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply

2

$2$ by

35

$35$ .

A = arccos (\frac{2450}{70 \cdot 37})

$A = \arccos (\frac{2450}{70 \cdot 37})$

Step 4.2.2

Multiply

70

$70$ by

37

$37$ .

A = arccos (\frac{2450}{2590})

$A = \arccos (\frac{2450}{2590})$

A = arccos (\frac{2450}{2590})

$A = \arccos (\frac{2450}{2590})$

Step 4.3

Cancel the common factor of

2450

$2450$ and

2590

$2590$ .

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Step 4.3.1

Factor

70

$70$ out of

2450

$2450$ .

A = arccos (\frac{70 (35)}{2590})

$A = \arccos (\frac{70 (35)}{2590})$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor

70

$70$ out of

2590

$2590$ .

A = arccos (\frac{70 \cdot 35}{70 \cdot 37})

$A = \arccos (\frac{70 \cdot 35}{70 \cdot 37})$

Step 4.3.2.2

Cancel the common factor.

$A = \arccos (\frac{70 \cdot 35}{70 \cdot 37})$

Step 4.3.2.3

Rewrite the expression.

$A = \arccos (\frac{35}{37})$

Step 4.4

Evaluate

$\arccos (\frac{35}{37})$ .

$A = 18.92464441$

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.

$B = \arccos (\frac{{(12)}^{2} + {(37)}^{2} - {(35)}^{2}}{2 (12) (37)})$

Step 8

Simplify the results.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise

$12$ to the power of

$2$ .

$B = \arccos (\frac{144 + 37^{2} - 35^{2}}{2 (12) \cdot 37})$

Step 8.1.2

Raise

$37$ to the power of

$2$ .

$B = \arccos (\frac{144 + 1369 - 35^{2}}{2 (12) \cdot 37})$

Step 8.1.3

Raise

$35$ to the power of

$2$ .

$B = \arccos (\frac{144 + 1369 - 1 \cdot 1225}{2 (12) \cdot 37})$

Step 8.1.4

Multiply

$- 1$ by

$1225$ .

$B = \arccos (\frac{144 + 1369 - 1225}{2 (12) \cdot 37})$

Step 8.1.5

Add

$144$ and

$1369$ .

$B = \arccos (\frac{1513 - 1225}{2 (12) \cdot 37})$

Step 8.1.6

Subtract

$1225$ from

$1513$ .

$B = \arccos (\frac{288}{2 (12) \cdot 37})$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply

$2$ by

$12$ .

$B = \arccos (\frac{288}{24 \cdot 37})$

Step 8.2.2

Multiply

$24$ by

$37$ .

$B = \arccos (\frac{288}{888})$

Step 8.3

Cancel the common factor of

$288$ and

$888$ .

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Step 8.3.1

Factor

$24$ out of

$288$ .

$B = \arccos (\frac{24 (12)}{888})$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor

$24$ out of

$888$ .

$B = \arccos (\frac{24 \cdot 12}{24 \cdot 37})$

Step 8.3.2.2

Cancel the common factor.

$B = \arccos (\frac{24 \cdot 12}{24 \cdot 37})$

Step 8.3.2.3

Rewrite the expression.

$B = \arccos (\frac{12}{37})$

Step 8.4

Evaluate

$\arccos (\frac{12}{37})$ .

$B = 71.07535558$

Step 9

The sum of all the angles in a triangle is

$180$ degrees.

$18.92464441 + C + 71.07535558 = 180$

Step 10

Solve the equation for

$C$ .

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Step 10.1

Add

$18.92464441$ and

$71.07535558$ .

$C + 90 = 180$

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.

$A = 18.92464441$

$B = 71.07535558$

$C = 90$

$a = 12$

$b = 35$

$c = 37$