Solve the Triangle a=12 , b=11 , A=35

$a = 12$ , $b = 11$ , $A = 35$

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 $B$ .

$\frac{\sin (B)}{11} = \frac{\sin (35)}{12}$

Solve the equation for $B$ .

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Simplify $\frac{\sin (35)}{12}$ .

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Evaluate $\sin (35)$ .

$\frac{\sin (B)}{11} = \frac{0.57357643}{12}$

Divide $0.57357643$ by $12$ .

$\frac{\sin (B)}{11} = 0.04779803$

Multiply both sides of the equation by $11$ .

$11 \cdot \frac{\sin (B)}{11} = 11 \cdot 0.04779803$

Simplify both sides of the equation.

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Cancel the common factor of $11$ .

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Cancel the common factor.

$11 \cdot \frac{\sin (B)}{11} = 11 \cdot 0.04779803$

Rewrite the expression.

$\sin (B) = 11 \cdot 0.04779803$

Multiply $11$ by $0.04779803$ .

$\sin (B) = 0.52577839$

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

$B = \arcsin (0.52577839)$

Evaluate $\arcsin (0.52577839)$ .

$B = 31.72065955$

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.

$B = 180 - 31.72065955$

Subtract $31.72065955$ from $180$ .

$B = 148.27934044$

The solution to the equation $B = 31.72065955$ .

$B = 31.72065955, 148.27934044$

Exclude the invalid angle.

$B = 31.72065955$

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

$35 + C + 31.72065955 = 180$

Solve the equation for $C$ .

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Add $35$ and $31.72065955$ .

$C + 66.72065955 = 180$

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

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Subtract $66.72065955$ from both sides of the equation.

$C = 180 - 66.72065955$

Subtract $66.72065955$ from $180$ .

$C = 113.27934044$

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

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

$\frac{\sin (113.27934044)}{c} = \frac{\sin (35)}{12}$

Solve the equation for $c$ .

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Factor each term.

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Evaluate $\sin (113.27934044)$ .

$\frac{0.91858894}{c} = \frac{\sin (35)}{12}$

Evaluate $\sin (35)$ .

$\frac{0.91858894}{c} = \frac{0.57357643}{12}$

Divide $0.57357643$ by $12$ .

$\frac{0.91858894}{c} = 0.04779803$

Solve for $c$ .

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Multiply each term by $c$ and simplify.

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Multiply each term in $\frac{0.91858894}{c} = 0.04779803$ by $c$ .

$\frac{0.91858894}{c} \cdot c = 0.04779803 \cdot c$

Cancel the common factor of $c$ .

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Cancel the common factor.

$\frac{0.91858894}{c} \cdot c = 0.04779803 \cdot c$

Rewrite the expression.

$0.91858894 = 0.04779803 \cdot c$

$0.91858894 = 0.04779803 c$

Rewrite the equation as $0.04779803 c = 0.91858894$ .

$0.04779803 c = 0.91858894$

Divide each term by $0.04779803$ and simplify.

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Divide each term in $0.04779803 c = 0.91858894$ by $0.04779803$ .

$\frac{0.04779803 c}{0.04779803} = \frac{0.91858894}{0.04779803}$

Cancel the common factor of $0.04779803$ .

$c = \frac{0.91858894}{0.04779803}$

Divide $0.91858894$ by $0.04779803$ .

$c = 19.21813145$

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

$A = 35$

$B = 31.72065955$

$C = 113.27934044$

$a = 12$

$b = 11$

$c = 19.21813145$

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