Solve the Triangle A=42 , a=26 , b=18

$A = 42$ , $a = 26$ , $b = 18$

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 $B$ .

$\frac{\sin (B)}{18} = \frac{\sin (42)}{26}$

Solve the equation for $B$ .

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Simplify $\frac{\sin (42)}{26}$ .

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Evaluate $\sin (42)$ .

$\frac{\sin (B)}{18} = \frac{0.6691306}{26}$

Divide $0.6691306$ by $26$ .

$\frac{\sin (B)}{18} = 0.02573579$

Multiply both sides of the equation by $18$ .

$18 \cdot \frac{\sin (B)}{18} = 18 \cdot 0.02573579$

Simplify both sides of the equation.

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Cancel the common factor of $18$ .

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Cancel the common factor.

$18 \cdot \frac{\sin (B)}{18} = 18 \cdot 0.02573579$

Rewrite the expression.

$\sin (B) = 18 \cdot 0.02573579$

Multiply $18$ by $0.02573579$ .

$\sin (B) = 0.46324426$

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

$B = \arcsin (0.46324426)$

Evaluate $\arcsin (0.46324426)$ .

$B = 27.59665301$

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.

$B = 180 - 27.59665301$

Subtract $27.59665301$ from $180$ .

$B = 152.40334698$

The solution to the equation $B = 27.59665301$ .

$B = 27.59665301, 152.40334698$

Exclude the invalid angle.

$B = 27.59665301$

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

$42 + C + 27.59665301 = 180$

Solve the equation for $C$ .

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Add $42$ and $27.59665301$ .

$C + 69.59665301 = 180$

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

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Subtract $69.59665301$ from both sides of the equation.

$C = 180 - 69.59665301$

Subtract $69.59665301$ from $180$ .

$C = 110.40334698$

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

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

$\frac{\sin (110.40334698)}{c} = \frac{\sin (42)}{26}$

Solve the equation for $c$ .

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Factor each term.

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Evaluate $\sin (110.40334698)$ .

$\frac{0.93726162}{c} = \frac{\sin (42)}{26}$

Evaluate $\sin (42)$ .

$\frac{0.93726162}{c} = \frac{0.6691306}{26}$

Divide $0.6691306$ by $26$ .

$\frac{0.93726162}{c} = 0.02573579$

Solve for $c$ .

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Multiply each term by $c$ and simplify.

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Multiply each term in $\frac{0.93726162}{c} = 0.02573579$ by $c$ .

$\frac{0.93726162}{c} \cdot c = 0.02573579 \cdot c$

Cancel the common factor of $c$ .

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Cancel the common factor.

$\frac{0.93726162}{c} \cdot c = 0.02573579 \cdot c$

Rewrite the expression.

$0.93726162 = 0.02573579 \cdot c$

$0.93726162 = 0.02573579 c$

Rewrite the equation as $0.02573579 c = 0.93726162$ .

$0.02573579 c = 0.93726162$

Divide each term by $0.02573579$ and simplify.

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Divide each term in $0.02573579 c = 0.93726162$ by $0.02573579$ .

$\frac{0.02573579 c}{0.02573579} = \frac{0.93726162}{0.02573579}$

Cancel the common factor of $0.02573579$ .

$c = \frac{0.93726162}{0.02573579}$

Divide $0.93726162$ by $0.02573579$ .

$c = 36.41860353$

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

$A = 42$

$B = 27.59665301$

$C = 110.40334698$

$a = 26$

$b = 18$

$c = 36.41860353$

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