Trigonometry Examples

$a = 4$ , $b = 10$ , $c = 7$

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.

$A = \arccos (\frac{{(10)}^{2} + {(7)}^{2} - {(4)}^{2}}{2 (10) (7)})$

Step 4

Simplify the results.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$A = \arccos (\frac{100 + 7^{2} - 4^{2}}{2 (10) \cdot 7})$

Step 4.1.2

Raise $7$ to the power of $2$ .

$A = \arccos (\frac{100 + 49 - 4^{2}}{2 (10) \cdot 7})$

Step 4.1.3

Raise $4$ to the power of $2$ .

$A = \arccos (\frac{100 + 49 - 1 \cdot 16}{2 (10) \cdot 7})$

Step 4.1.4

Multiply $- 1$ by $16$ .

$A = \arccos (\frac{100 + 49 - 16}{2 (10) \cdot 7})$

Step 4.1.5

Add $100$ and $49$ .

$A = \arccos (\frac{149 - 16}{2 (10) \cdot 7})$

Step 4.1.6

Subtract $16$ from $149$ .

$A = \arccos (\frac{133}{2 (10) \cdot 7})$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $2$ by $10$ .

$A = \arccos (\frac{133}{20 \cdot 7})$

Step 4.2.2

Multiply $20$ by $7$ .

$A = \arccos (\frac{133}{140})$

Step 4.3

Cancel the common factor of $133$ and $140$ .

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Step 4.3.1

Factor $7$ out of $133$ .

$A = \arccos (\frac{7 (19)}{140})$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $7$ out of $140$ .

$A = \arccos (\frac{7 \cdot 19}{7 \cdot 20})$

Step 4.3.2.2

Cancel the common factor.

$A = \arccos (\frac{7 \cdot 19}{7 \cdot 20})$

Step 4.3.2.3

Rewrite the expression.

$A = \arccos (\frac{19}{20})$

Step 4.4

Evaluate $\arccos (\frac{19}{20})$ .

$A = 18.19487233$

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.

$B = \arccos (\frac{{(4)}^{2} + {(7)}^{2} - {(10)}^{2}}{2 (4) (7)})$

Step 8

Simplify the results.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $4$ to the power of $2$ .

$B = \arccos (\frac{16 + 7^{2} - 10^{2}}{2 (4) \cdot 7})$

Step 8.1.2

Raise $7$ to the power of $2$ .

$B = \arccos (\frac{16 + 49 - 10^{2}}{2 (4) \cdot 7})$

Step 8.1.3

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

$B = \arccos (\frac{16 + 49 - 1 \cdot 100}{2 (4) \cdot 7})$

Step 8.1.4

Multiply $- 1$ by $100$ .

$B = \arccos (\frac{16 + 49 - 100}{2 (4) \cdot 7})$

Step 8.1.5

Add $16$ and $49$ .

$B = \arccos (\frac{65 - 100}{2 (4) \cdot 7})$

Step 8.1.6

Subtract $100$ from $65$ .

$B = \arccos (\frac{- 35}{2 (4) \cdot 7})$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $2$ by $4$ .

$B = \arccos (\frac{- 35}{8 \cdot 7})$

Step 8.2.2

Multiply $8$ by $7$ .

$B = \arccos (\frac{- 35}{56})$

Step 8.3

Reduce the expression by cancelling the common factors.

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Step 8.3.1

Cancel the common factor of $- 35$ and $56$ .

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Step 8.3.1.1

Factor $7$ out of $- 35$ .

$B = \arccos (\frac{7 (- 5)}{56})$

Step 8.3.1.2

Cancel the common factors.

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Step 8.3.1.2.1

Factor $7$ out of $56$ .

$B = \arccos (\frac{7 \cdot - 5}{7 \cdot 8})$

Step 8.3.1.2.2

Cancel the common factor.

$B = \arccos (\frac{7 \cdot - 5}{7 \cdot 8})$

Step 8.3.1.2.3

Rewrite the expression.

$B = \arccos (\frac{- 5}{8})$

Step 8.3.2

Move the negative in front of the fraction.

$B = \arccos (- \frac{5}{8})$

Step 8.4

Evaluate $\arccos (- \frac{5}{8})$ .

$B = 128.68218745$

Step 9

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

$18.19487233 + C + 128.68218745 = 180$

Step 10

Solve the equation for $C$ .

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Step 10.1

Add $18.19487233$ and $128.68218745$ .

$C + 146.87705979 = 180$

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 $146.87705979$ from both sides of the equation.

$C = 180 - 146.87705979$

Step 10.2.2

Subtract $146.87705979$ from $180$ .

$C = 33.1229402$

Step 11

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

$A = 18.19487233$

$B = 128.68218745$

$C = 33.1229402$

$a = 4$

$b = 10$

$c = 7$