Solve the Triangle B=48 , a=31 , b=26

$B = 48$ , $a = 31$ , $b = 26$

The Law of Sines produces an ambiguous angle result. This means that there are $2$ angles that will correctly solve the equation. For the first triangle, use the first possible angle value.

Solve for the first triangle.

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 $A$ .

$\frac{\sin (A)}{31} = \frac{\sin (48)}{26}$

Solve the equation for $A$ .

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Simplify $\frac{\sin (48)}{26}$ .

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Evaluate $\sin (48)$ .

$\frac{\sin (A)}{31} = \frac{0.74314482}{26}$

Divide $0.74314482$ by $26$ .

$\frac{\sin (A)}{31} = 0.02858249$

Multiply both sides of the equation by $31$ .

$31 \cdot \frac{\sin (A)}{31} = 31 \cdot 0.02858249$

Simplify both sides of the equation.

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Cancel the common factor of $31$ .

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Cancel the common factor.

$31 \cdot \frac{\sin (A)}{31} = 31 \cdot 0.02858249$

Rewrite the expression.

$\sin (A) = 31 \cdot 0.02858249$

Multiply $31$ by $0.02858249$ .

$\sin (A) = 0.88605729$

Take the inverse sine of both sides of the equation to extract $A$ from inside the sine.

$A = \arcsin (0.88605729)$

Evaluate $\arcsin (0.88605729)$ .

$A = 62.38191412$

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$A = 180 - 62.38191412$

Subtract $62.38191412$ from $180$ .

$A = 117.61808587$

The solution to the equation $A = 62.38191412$ .

$A = 62.38191412, 117.61808587$

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

$62.38191412 + C + 48 = 180$

Solve the equation for $C$ .

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Add $62.38191412$ and $48$ .

$C + 110.38191412 = 180$

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

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Subtract $110.38191412$ from both sides of the equation.

$C = 180 - 110.38191412$

Subtract $110.38191412$ from $180$ .

$C = 69.61808587$

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

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

$\frac{\sin (69.61808587)}{c} = \frac{\sin (62.38191412)}{31}$

Solve the equation for $c$ .

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Factor each term.

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Evaluate $\sin (69.61808587)$ .

$\frac{0.93739197}{c} = \frac{\sin (62.38191412)}{31}$

Evaluate $\sin (62.38191412)$ .

$\frac{0.93739197}{c} = \frac{0.88605729}{31}$

Divide $0.88605729$ by $31$ .

$\frac{0.93739197}{c} = 0.02858249$

Solve for $c$ .

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Multiply each term by $c$ and simplify.

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Multiply each term in $\frac{0.93739197}{c} = 0.02858249$ by $c$ .

$\frac{0.93739197}{c} \cdot c = 0.02858249 \cdot c$

Cancel the common factor of $c$ .

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Cancel the common factor.

$\frac{0.93739197}{c} \cdot c = 0.02858249 \cdot c$

Rewrite the expression.

$0.93739197 = 0.02858249 \cdot c$

$0.93739197 = 0.02858249 c$

Rewrite the equation as $0.02858249 c = 0.93739197$ .

$0.02858249 c = 0.93739197$

Divide each term by $0.02858249$ and simplify.

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Divide each term in $0.02858249 c = 0.93739197$ by $0.02858249$ .

$\frac{0.02858249 c}{0.02858249} = \frac{0.93739197}{0.02858249}$

Cancel the common factor of $0.02858249$ .

$c = \frac{0.93739197}{0.02858249}$

Divide $0.93739197$ by $0.02858249$ .

$c = 32.79601828$

For the second triangle, use the second possible angle value.

Solve for the second triangle.

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

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

$\frac{\sin (A)}{31} = \frac{\sin (48)}{26}$

Solve the equation for $A$ .

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Simplify $\frac{\sin (48)}{26}$ .

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Evaluate $\sin (48)$ .

$\frac{\sin (A)}{31} = \frac{0.74314482}{26}$

Divide $0.74314482$ by $26$ .

$\frac{\sin (A)}{31} = 0.02858249$

Multiply both sides of the equation by $31$ .

$31 \cdot \frac{\sin (A)}{31} = 31 \cdot 0.02858249$

Simplify both sides of the equation.

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Cancel the common factor of $31$ .

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Cancel the common factor.

$31 \cdot \frac{\sin (A)}{31} = 31 \cdot 0.02858249$

Rewrite the expression.

$\sin (A) = 31 \cdot 0.02858249$

Multiply $31$ by $0.02858249$ .

$\sin (A) = 0.88605729$

Take the inverse sine of both sides of the equation to extract $A$ from inside the sine.

$A = \arcsin (0.88605729)$

Evaluate $\arcsin (0.88605729)$ .

$A = 62.38191412$

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$A = 180 - 62.38191412$

Subtract $62.38191412$ from $180$ .

$A = 117.61808587$

The solution to the equation $A = 62.38191412$ .

$A = 62.38191412, 117.61808587$

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

$117.61808587 + C + 48 = 180$

Solve the equation for $C$ .

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Add $117.61808587$ and $48$ .

$C + 165.61808587 = 180$

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

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Subtract $165.61808587$ from both sides of the equation.

$C = 180 - 165.61808587$

Subtract $165.61808587$ from $180$ .

$C = 14.38191412$

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

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

$\frac{\sin (14.38191412)}{c} = \frac{\sin (117.61808587)}{31}$

Solve the equation for $c$ .

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Factor each term.

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Evaluate $\sin (14.38191412)$ .

$\frac{0.24838413}{c} = \frac{\sin (117.61808587)}{31}$

Evaluate $\sin (117.61808587)$ .

$\frac{0.24838413}{c} = \frac{0.88605729}{31}$

Divide $0.88605729$ by $31$ .

$\frac{0.24838413}{c} = 0.02858249$

Solve for $c$ .

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Multiply each term by $c$ and simplify.

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Multiply each term in $\frac{0.24838413}{c} = 0.02858249$ by $c$ .

$\frac{0.24838413}{c} \cdot c = 0.02858249 \cdot c$

Cancel the common factor of $c$ .

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Cancel the common factor.

$\frac{0.24838413}{c} \cdot c = 0.02858249 \cdot c$

Rewrite the expression.

$0.24838413 = 0.02858249 \cdot c$

$0.24838413 = 0.02858249 c$

Rewrite the equation as $0.02858249 c = 0.24838413$ .

$0.02858249 c = 0.24838413$

Divide each term by $0.02858249$ and simplify.

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Divide each term in $0.02858249 c = 0.24838413$ by $0.02858249$ .

$\frac{0.02858249 c}{0.02858249} = \frac{0.24838413}{0.02858249}$

Cancel the common factor of $0.02858249$ .

$c = \frac{0.24838413}{0.02858249}$

Divide $0.24838413$ by $0.02858249$ .

$c = 8.69007931$

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

First Triangle Combination:

$A = 62.38191412$

$B = 48$

$C = 69.61808587$

$a = 31$

$b = 26$

$c = 32.79601828$

Second Triangle Combination:

$A = 117.61808587$

$B = 48$

$C = 14.38191412$

$a = 31$

$b = 26$

$c = 8.69007931$

Solve the Triangle B=48 , a=31 , b=26

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