Precalculus Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = \\ a = & 12 \end{matrix} & \begin{matrix} A = & 102 ° \\ B = \\ C = & 40 ° \end{matrix} \end{matrix}$

Step 1

The law of sines is based on the proportionality of sides and angles in triangles. The law states that for the angles of a non-right triangle, each angle of the triangle has the same ratio of angle measure to sine value.

$\frac{\sin (A)}{a} = \frac{\sin (B)}{b} = \frac{\sin (C)}{c}$

Step 2

Substitute the known values into the law of sines to find $c$ .

$\frac{\sin (40 °)}{c} = \frac{\sin (102 °)}{12}$

Step 3

Solve the equation for $c$ .

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Step 3.1

Factor each term.

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Step 3.1.1

Evaluate $\sin (40 °)$ .

$\frac{0.6427876}{c} = \frac{\sin (102 °)}{12}$

Step 3.1.2

Evaluate $\sin (102 °)$ .

$\frac{0.6427876}{c} = \frac{0.9781476}{12}$

Step 3.1.3

Divide $0.9781476$ by $12$ .

$\frac{0.6427876}{c} = 0.0815123$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$c, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$c$

Step 3.3

Multiply each term in $\frac{0.6427876}{c} = 0.0815123$ by $c$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{0.6427876}{c} = 0.0815123$ by $c$ .

$\frac{0.6427876}{c} c = 0.0815123 c$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $c$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{0.6427876}{c} c = 0.0815123 c$

Step 3.3.2.1.2

Rewrite the expression.

$0.6427876 = 0.0815123 c$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $0.0815123 c = 0.6427876$ .

$0.0815123 c = 0.6427876$

Step 3.4.2

Divide each term in $0.0815123 c = 0.6427876$ by $0.0815123$ and simplify.

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Step 3.4.2.1

Divide each term in $0.0815123 c = 0.6427876$ by $0.0815123$ .

$\frac{0.0815123 c}{0.0815123} = \frac{0.6427876}{0.0815123}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $0.0815123$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{0.0815123 c}{0.0815123} = \frac{0.6427876}{0.0815123}$

Step 3.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.6427876}{0.0815123}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $0.6427876$ by $0.0815123$ .

$c = 7.8857744$

Step 4

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

$102 ° + 40 ° + B = 180$

Step 5

Solve the equation for $B$ .

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Step 5.1

Add $102 °$ and $40 °$ .

$142 + B = 180$

Step 5.2

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

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Step 5.2.1

Subtract $142$ from both sides of the equation.

$B = 180 - 142$

Step 5.2.2

Subtract $142$ from $180$ .

$B = 38$

Step 6

$\frac{\sin (A)}{a} = \frac{\sin (B)}{b} = \frac{\sin (C)}{c}$

Step 7

Substitute the known values into the law of sines to find $b$ .

$\frac{\sin (38)}{b} = \frac{\sin (102 °)}{12}$

Step 8

Solve the equation for $b$ .

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Step 8.1

Factor each term.

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Step 8.1.1

Evaluate $\sin (38)$ .

$\frac{0.61566147}{b} = \frac{\sin (102 °)}{12}$

Step 8.1.2

Evaluate $\sin (102 °)$ .

$\frac{0.61566147}{b} = \frac{0.9781476}{12}$

Step 8.1.3

Divide $0.9781476$ by $12$ .

$\frac{0.61566147}{b} = 0.0815123$

Step 8.2

Find the LCD of the terms in the equation.

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Step 8.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$b, 1$

Step 8.2.2

The LCM of one and any expression is the expression.

$b$

Step 8.3

Multiply each term in $\frac{0.61566147}{b} = 0.0815123$ by $b$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in $\frac{0.61566147}{b} = 0.0815123$ by $b$ .

$\frac{0.61566147}{b} b = 0.0815123 b$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of $b$ .

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Step 8.3.2.1.1

Cancel the common factor.

$\frac{0.61566147}{b} b = 0.0815123 b$

Step 8.3.2.1.2

Rewrite the expression.

$0.61566147 = 0.0815123 b$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as $0.0815123 b = 0.61566147$ .

$0.0815123 b = 0.61566147$

Step 8.4.2

Divide each term in $0.0815123 b = 0.61566147$ by $0.0815123$ and simplify.

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Step 8.4.2.1

Divide each term in $0.0815123 b = 0.61566147$ by $0.0815123$ .

$\frac{0.0815123 b}{0.0815123} = \frac{0.61566147}{0.0815123}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $0.0815123$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.0815123 b}{0.0815123} = \frac{0.61566147}{0.0815123}$

Step 8.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{0.61566147}{0.0815123}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide $0.61566147$ by $0.0815123$ .

$b = 7.55298862$

Step 9

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

$A = 102 °$

$B = 38$

$C = 40 °$

$a = 12$

$b = 7.55298862$

$c = 7.8857744$