Solve the Triangle B=30 degrees , C=43 degrees ; and a=8

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Trigonometry

$B = 30 °$ , $C = 43 °$ ; and $a = 8$

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

$A + 43 ° + 30 ° = 180$

Solve the equation for $A$ .

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Add $43 °$ and $30 °$ .

$A + 73 = 180$

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

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Subtract $73$ from both sides of the equation.

$A = 180 - 73$

Subtract $73$ from $180$ .

$A = 107$

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.

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

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

$\frac{\sin (30 °)}{b} = \frac{\sin (107)}{8}$

Solve the equation for $b$ .

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Factor each term.

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The exact value of $\sin (30 °)$ is $\frac{1}{2}$ .

$\frac{\frac{1}{2}}{b} = \frac{\sin (107)}{8}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} \cdot \frac{1}{b} = \frac{\sin (107)}{8}$

Multiply $\frac{1}{2}$ and $\frac{1}{b}$ .

$\frac{1}{2 b} = \frac{\sin (107)}{8}$

Evaluate $\sin (107)$ .

$\frac{1}{2 b} = \frac{0.95630475}{8}$

Divide $0.95630475$ by $8$ .

$\frac{1}{2 b} = 0.11953809$

Find the LCD of the terms in the equation.

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

$2 b, 1$

The LCM of one and any expression is the expression.

$2 b$

Multiply each term in $\frac{1}{2 b} = 0.11953809$ by $2 b$ to eliminate the fractions.

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Multiply each term in $\frac{1}{2 b} = 0.11953809$ by $2 b$ .

$\frac{1}{2 b} (2 b) = 0.11953809 (2 b)$

Simplify the left side.

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Rewrite using the commutative property of multiplication.

$2 \frac{1}{2 b} b = 0.11953809 (2 b)$

Cancel the common factor of $2$ .

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Factor $2$ out of $2 b$ .

$2 \frac{1}{2 (b)} b = 0.11953809 (2 b)$

Cancel the common factor.

$2 \frac{1}{2 b} b = 0.11953809 (2 b)$

Rewrite the expression.

$\frac{1}{b} b = 0.11953809 (2 b)$

Cancel the common factor of $b$ .

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Cancel the common factor.

$\frac{1}{b} b = 0.11953809 (2 b)$

Rewrite the expression.

$1 = 0.11953809 (2 b)$

Simplify the right side.

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Multiply $2$ by $0.11953809$ .

$1 = 0.23907618 b$

Solve the equation.

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Rewrite the equation as $0.23907618 b = 1$ .

$0.23907618 b = 1$

Divide each term in $0.23907618 b = 1$ by $0.23907618$ and simplify.

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Divide each term in $0.23907618 b = 1$ by $0.23907618$ .

$\frac{0.23907618 b}{0.23907618} = \frac{1}{0.23907618}$

Simplify the left side.

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Cancel the common factor of $0.23907618$ .

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Cancel the common factor.

$\frac{0.23907618 b}{0.23907618} = \frac{1}{0.23907618}$

Divide $b$ by $1$ .

$b = \frac{1}{0.23907618}$

Simplify the right side.

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Divide $1$ by $0.23907618$ .

$b = 4.18276702$

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

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

$\frac{\sin (43 °)}{c} = \frac{\sin (107)}{8}$

Solve the equation for $c$ .

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Factor each term.

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Evaluate $\sin (43 °)$ .

$\frac{0.68199836}{c} = \frac{\sin (107)}{8}$

Evaluate $\sin (107)$ .

$\frac{0.68199836}{c} = \frac{0.95630475}{8}$

Divide $0.95630475$ by $8$ .

$\frac{0.68199836}{c} = 0.11953809$

Find the LCD of the terms in the equation.

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

$c, 1$

The LCM of one and any expression is the expression.

$c$

Multiply each term in $\frac{0.68199836}{c} = 0.11953809$ by $c$ to eliminate the fractions.

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Multiply each term in $\frac{0.68199836}{c} = 0.11953809$ by $c$ .

$\frac{0.68199836}{c} c = 0.11953809 c$

Simplify the left side.

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Cancel the common factor of $c$ .

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Cancel the common factor.

$\frac{0.68199836}{c} c = 0.11953809 c$

Rewrite the expression.

$0.68199836 = 0.11953809 c$

Solve the equation.

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Rewrite the equation as $0.11953809 c = 0.68199836$ .

$0.11953809 c = 0.68199836$

Divide each term in $0.11953809 c = 0.68199836$ by $0.11953809$ and simplify.

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Divide each term in $0.11953809 c = 0.68199836$ by $0.11953809$ .

$\frac{0.11953809 c}{0.11953809} = \frac{0.68199836}{0.11953809}$

Simplify the left side.

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Cancel the common factor of $0.11953809$ .

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Cancel the common factor.

$\frac{0.11953809 c}{0.11953809} = \frac{0.68199836}{0.11953809}$

Divide $c$ by $1$ .

$c = \frac{0.68199836}{0.11953809}$

Simplify the right side.

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Divide $0.68199836$ by $0.11953809$ .

$c = 5.7052805$

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

$A = 107$

$B = 30 °$

$C = 43 °$

$a = 8$

$b = 4.18276702$

$c = 5.7052805$

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