Trigonometry Examples

$a = 10$ ,

$b = 9$ ,

$c = 7$

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.

$A = \arccos (\frac{{(9)}^{2} + {(7)}^{2} - {(10)}^{2}}{2 (9) (7)})$

Step 4

Simplify the results.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise

$9$ to the power of

$2$ .

$A = \arccos (\frac{81 + 7^{2} - 10^{2}}{2 (9) \cdot 7})$

Step 4.1.2

Raise

$7$ to the power of

$2$ .

$A = \arccos (\frac{81 + 49 - 10^{2}}{2 (9) \cdot 7})$

Step 4.1.3

Raise

$10$ to the power of

$2$ .

$A = \arccos (\frac{81 + 49 - 1 \cdot 100}{2 (9) \cdot 7})$

Step 4.1.4

Multiply

$- 1$ by

$100$ .

$A = \arccos (\frac{81 + 49 - 100}{2 (9) \cdot 7})$

Step 4.1.5

Add

$81$ and

$49$ .

$A = \arccos (\frac{130 - 100}{2 (9) \cdot 7})$

Step 4.1.6

Subtract

$100$ from

$130$ .

$A = \arccos (\frac{30}{2 (9) \cdot 7})$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply

$2$ by

$9$ .

$A = \arccos (\frac{30}{18 \cdot 7})$

Step 4.2.2

Multiply

$18$ by

$7$ .

$A = \arccos (\frac{30}{126})$

Step 4.3

Cancel the common factor of

$30$ and

$126$ .

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Step 4.3.1

Factor

$6$ out of

$30$ .

$A = \arccos (\frac{6 (5)}{126})$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor

$6$ out of

$126$ .

$A = \arccos (\frac{6 \cdot 5}{6 \cdot 21})$

Step 4.3.2.2

Cancel the common factor.

$A = \arccos (\frac{6 \cdot 5}{6 \cdot 21})$

Step 4.3.2.3

Rewrite the expression.

$A = \arccos (\frac{5}{21})$

Step 4.4

Evaluate

$\arccos (\frac{5}{21})$ .

$A = 76.225853$

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.

$B = \arccos (\frac{{(10)}^{2} + {(7)}^{2} - {(9)}^{2}}{2 (10) (7)})$

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

$2$ .

$B = \arccos (\frac{100 + 7^{2} - 9^{2}}{2 (10) \cdot 7})$

Step 8.1.2

Raise

$7$ to the power of

$2$ .

$B = \arccos (\frac{100 + 49 - 9^{2}}{2 (10) \cdot 7})$

Step 8.1.3

Raise

$9$ to the power of

$2$ .

$B = \arccos (\frac{100 + 49 - 1 \cdot 81}{2 (10) \cdot 7})$

Step 8.1.4

Multiply

$- 1$ by

$81$ .

$B = \arccos (\frac{100 + 49 - 81}{2 (10) \cdot 7})$

Step 8.1.5

Add

$100$ and

$49$ .

$B = \arccos (\frac{149 - 81}{2 (10) \cdot 7})$

Step 8.1.6

Subtract

$81$ from

$149$ .

$B = \arccos (\frac{68}{2 (10) \cdot 7})$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply

$2$ by

$10$ .

$B = \arccos (\frac{68}{20 \cdot 7})$

Step 8.2.2

Multiply

$20$ by

$7$ .

$B = \arccos (\frac{68}{140})$

Step 8.3

Cancel the common factor of

$68$ and

$140$ .

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Step 8.3.1

Factor

$4$ out of

$68$ .

$B = \arccos (\frac{4 (17)}{140})$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor

$4$ out of

$140$ .

$B = \arccos (\frac{4 \cdot 17}{4 \cdot 35})$

Step 8.3.2.2

Cancel the common factor.

$B = \arccos (\frac{4 \cdot 17}{4 \cdot 35})$

Step 8.3.2.3

Rewrite the expression.

$B = \arccos (\frac{17}{35})$

Step 8.4

Evaluate

$\arccos (\frac{17}{35})$ .

$B = 60.94071893$

Step 9

The sum of all the angles in a triangle is

$180$ degrees.

$76.225853 + C + 60.94071893 = 180$

Step 10

Solve the equation for

$C$ .

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Step 10.1

Add

$76.225853$ and

$60.94071893$ .

$C + 137.16657193 = 180$

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

$137.16657193$ from both sides of the equation.

$C = 180 - 137.16657193$

Step 10.2.2

Subtract

$137.16657193$ from

$180$ .

$C = 42.83342806$

Step 11

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

$A = 76.225853$

$B = 60.94071893$

$C = 42.83342806$

$a = 10$

$b = 9$

$c = 7$