Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 1 \\ c = & 1 \\ a = & 1 \end{matrix} & \begin{matrix} A = \\ B = \\ C = \end{matrix} \end{matrix}$

Step 1

Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.

$a^{2} = b^{2} + c^{2} - 2 b c \cos (A)$

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.

$A = \arccos (\frac{{(1)}^{2} + {(1)}^{2} - {(1)}^{2}}{2 (1) (1)})$

Step 4

Simplify the results.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

One to any power is one.

$A = \arccos (\frac{1 + 1^{2} - 1^{2}}{2 (1) \cdot 1})$

Step 4.1.2

One to any power is one.

$A = \arccos (\frac{1 + 1 - 1^{2}}{2 (1) \cdot 1})$

Step 4.1.3

One to any power is one.

$A = \arccos (\frac{1 + 1 - 1 \cdot 1}{2 (1) \cdot 1})$

Step 4.1.4

Multiply $- 1$ by $1$ .

$A = \arccos (\frac{1 + 1 - 1}{2 (1) \cdot 1})$

Step 4.1.5

Add $1$ and $1$ .

$A = \arccos (\frac{2 - 1}{2 (1) \cdot 1})$

Step 4.1.6

Subtract $1$ from $2$ .

$A = \arccos (\frac{1}{2 (1) \cdot 1})$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $2$ by $1$ .

$A = \arccos (\frac{1}{2 \cdot 1})$

Step 4.2.2

Multiply $2$ by $1$ .

$A = \arccos (\frac{1}{2})$

Step 4.3

The exact value of $\arccos (\frac{1}{2})$ is $60$ .

$A = 60$

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.

$B = \arccos (\frac{{(1)}^{2} + {(1)}^{2} - {(1)}^{2}}{2 (1) (1)})$

Step 8

Simplify the results.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

One to any power is one.

$B = \arccos (\frac{1 + 1^{2} - 1^{2}}{2 (1) \cdot 1})$

Step 8.1.2

One to any power is one.

$B = \arccos (\frac{1 + 1 - 1^{2}}{2 (1) \cdot 1})$

Step 8.1.3

One to any power is one.

$B = \arccos (\frac{1 + 1 - 1 \cdot 1}{2 (1) \cdot 1})$

Step 8.1.4

Multiply $- 1$ by $1$ .

$B = \arccos (\frac{1 + 1 - 1}{2 (1) \cdot 1})$

Step 8.1.5

Add $1$ and $1$ .

$B = \arccos (\frac{2 - 1}{2 (1) \cdot 1})$

Step 8.1.6

Subtract $1$ from $2$ .

$B = \arccos (\frac{1}{2 (1) \cdot 1})$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $2$ by $1$ .

$B = \arccos (\frac{1}{2 \cdot 1})$

Step 8.2.2

Multiply $2$ by $1$ .

$B = \arccos (\frac{1}{2})$

Step 8.3

The exact value of $\arccos (\frac{1}{2})$ is $60$ .

$B = 60$

Step 9

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

$60 + C + 60 = 180$

Step 10

Solve the equation for $C$ .

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Step 10.1

Add $60$ and $60$ .

$C + 120 = 180$

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 $120$ from both sides of the equation.

$C = 180 - 120$

Step 10.2.2

Subtract $120$ from $180$ .

$C = 60$

Step 11

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

$A = 60$

$B = 60$

$C = 60$

$a = 1$

$b = 1$

$c = 1$