Trigonometry Examples

$a = 4$ , $b = 7$ , $c = 6$

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.

$A = \arccos (\frac{{(7)}^{2} + {(6)}^{2} - {(4)}^{2}}{2 (7) (6)})$

Step 4

Simplify the results.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

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

$A = \arccos (\frac{49 + 6^{2} - 4^{2}}{2 (7) \cdot 6})$

Step 4.1.2

Raise $6$ to the power of $2$ .

$A = \arccos (\frac{49 + 36 - 4^{2}}{2 (7) \cdot 6})$

Step 4.1.3

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

$A = \arccos (\frac{49 + 36 - 1 \cdot 16}{2 (7) \cdot 6})$

Step 4.1.4

Multiply $- 1$ by $16$ .

$A = \arccos (\frac{49 + 36 - 16}{2 (7) \cdot 6})$

Step 4.1.5

Add $49$ and $36$ .

$A = \arccos (\frac{85 - 16}{2 (7) \cdot 6})$

Step 4.1.6

Subtract $16$ from $85$ .

$A = \arccos (\frac{69}{2 (7) \cdot 6})$

Step 4.2

Simplify the denominator.

Tap for more steps...

Step 4.2.1

Multiply $2$ by $7$ .

$A = \arccos (\frac{69}{14 \cdot 6})$

Step 4.2.2

Multiply $14$ by $6$ .

$A = \arccos (\frac{69}{84})$

Step 4.3

Cancel the common factor of $69$ and $84$ .

Tap for more steps...

Step 4.3.1

Factor $3$ out of $69$ .

$A = \arccos (\frac{3 (23)}{84})$

Step 4.3.2

Cancel the common factors.

Tap for more steps...

Step 4.3.2.1

Factor $3$ out of $84$ .

$A = \arccos (\frac{3 \cdot 23}{3 \cdot 28})$

Step 4.3.2.2

Cancel the common factor.

$A = \arccos (\frac{3 \cdot 23}{3 \cdot 28})$

Step 4.3.2.3

Rewrite the expression.

$A = \arccos (\frac{23}{28})$

Step 4.4

Evaluate $\arccos (\frac{23}{28})$ .

$A = 34.77194403$

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.

$B = \arccos (\frac{{(4)}^{2} + {(6)}^{2} - {(7)}^{2}}{2 (4) (6)})$

Step 8

Simplify the results.

Tap for more steps...

Step 8.1

Simplify the numerator.

Tap for more steps...

Step 8.1.1

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

$B = \arccos (\frac{16 + 6^{2} - 7^{2}}{2 (4) \cdot 6})$

Step 8.1.2

Raise $6$ to the power of $2$ .

$B = \arccos (\frac{16 + 36 - 7^{2}}{2 (4) \cdot 6})$

Step 8.1.3

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

$B = \arccos (\frac{16 + 36 - 1 \cdot 49}{2 (4) \cdot 6})$

Step 8.1.4

Multiply $- 1$ by $49$ .

$B = \arccos (\frac{16 + 36 - 49}{2 (4) \cdot 6})$

Step 8.1.5

Add $16$ and $36$ .

$B = \arccos (\frac{52 - 49}{2 (4) \cdot 6})$

Step 8.1.6

Subtract $49$ from $52$ .

$B = \arccos (\frac{3}{2 (4) \cdot 6})$

Step 8.2

Simplify the denominator.

Tap for more steps...

Step 8.2.1

Multiply $2$ by $4$ .

$B = \arccos (\frac{3}{8 \cdot 6})$

Step 8.2.2

Multiply $8$ by $6$ .

$B = \arccos (\frac{3}{48})$

Step 8.3

Cancel the common factor of $3$ and $48$ .

Tap for more steps...

Step 8.3.1

Factor $3$ out of $3$ .

$B = \arccos (\frac{3 (1)}{48})$

Step 8.3.2

Cancel the common factors.

Tap for more steps...

Step 8.3.2.1

Factor $3$ out of $48$ .

$B = \arccos (\frac{3 \cdot 1}{3 \cdot 16})$

Step 8.3.2.2

Cancel the common factor.

$B = \arccos (\frac{3 \cdot 1}{3 \cdot 16})$

Step 8.3.2.3

Rewrite the expression.

$B = \arccos (\frac{1}{16})$

Step 8.4

Evaluate $\arccos (\frac{1}{16})$ .

$B = 86.4166783$

Step 9

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

$34.77194403 + C + 86.4166783 = 180$

Step 10

Solve the equation for $C$ .

Tap for more steps...

Step 10.1

Add $34.77194403$ and $86.4166783$ .

$C + 121.18862233 = 180$

Step 10.2

Move all terms not containing $C$ to the right side of the equation.

Tap for more steps...

Step 10.2.1

Subtract $121.18862233$ from both sides of the equation.

$C = 180 - 121.18862233$

Step 10.2.2

Subtract $121.18862233$ from $180$ .

$C = 58.81137766$

Step 11

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

$A = 34.77194403$

$B = 86.4166783$

$C = 58.81137766$

$a = 4$

$b = 7$

$c = 6$