Solve the Triangle A=33 , a=20 , b=13

$A = 33$ , $a = 20$ , $b = 13$

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 $B$ .

$\frac{\sin (B)}{13} = \frac{\sin (33)}{20}$

Solve the equation for $B$ .

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Simplify $\frac{\sin (33)}{20}$ .

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Evaluate $\sin (33)$ .

$\frac{\sin (B)}{13} = \frac{0.54463903}{20}$

Divide $0.54463903$ by $20$ .

$\frac{\sin (B)}{13} = 0.02723195$

Multiply both sides of the equation by $13$ .

$13 \cdot \frac{\sin (B)}{13} = 13 \cdot 0.02723195$

Simplify both sides of the equation.

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Cancel the common factor of $13$ .

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Cancel the common factor.

$13 \cdot \frac{\sin (B)}{13} = 13 \cdot 0.02723195$

Rewrite the expression.

$\sin (B) = 13 \cdot 0.02723195$

Multiply $13$ by $0.02723195$ .

$\sin (B) = 0.35401537$

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

$B = \arcsin (0.35401537)$

Evaluate $\arcsin (0.35401537)$ .

$B = 20.73311092$

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.

$B = 180 - 20.73311092$

Subtract $20.73311092$ from $180$ .

$B = 159.26688907$

The solution to the equation $B = 20.73311092$ .

$B = 20.73311092, 159.26688907$

Exclude the invalid angle.

$B = 20.73311092$

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

$33 + C + 20.73311092 = 180$

Solve the equation for $C$ .

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Add $33$ and $20.73311092$ .

$C + 53.73311092 = 180$

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

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Subtract $53.73311092$ from both sides of the equation.

$C = 180 - 53.73311092$

Subtract $53.73311092$ from $180$ .

$C = 126.26688907$

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

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

$\frac{\sin (126.26688907)}{c} = \frac{\sin (33)}{20}$

Solve the equation for $c$ .

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Factor each term.

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Evaluate $\sin (126.26688907)$ .

$\frac{0.80627026}{c} = \frac{\sin (33)}{20}$

Evaluate $\sin (33)$ .

$\frac{0.80627026}{c} = \frac{0.54463903}{20}$

Divide $0.54463903$ by $20$ .

$\frac{0.80627026}{c} = 0.02723195$

Solve for $c$ .

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Multiply each term by $c$ and simplify.

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Multiply each term in $\frac{0.80627026}{c} = 0.02723195$ by $c$ .

$\frac{0.80627026}{c} \cdot c = 0.02723195 \cdot c$

Cancel the common factor of $c$ .

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Cancel the common factor.

$\frac{0.80627026}{c} \cdot c = 0.02723195 \cdot c$

Rewrite the expression.

$0.80627026 = 0.02723195 \cdot c$

$0.80627026 = 0.02723195 c$

Rewrite the equation as $0.02723195 c = 0.80627026$ .

$0.02723195 c = 0.80627026$

Divide each term by $0.02723195$ and simplify.

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Divide each term in $0.02723195 c = 0.80627026$ by $0.02723195$ .

$\frac{0.02723195 c}{0.02723195} = \frac{0.80627026}{0.02723195}$

Cancel the common factor of $0.02723195$ .

$c = \frac{0.80627026}{0.02723195}$

Divide $0.80627026$ by $0.02723195$ .

$c = 29.60750945$

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

$A = 33$

$B = 20.73311092$

$C = 126.26688907$

$a = 20$

$b = 13$

$c = 29.60750945$

Solve the Triangle A=33 , a=20 , b=13

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