# Law of Sines Calculator

## Solve triangles using the law of sines

The calculator will solve the given triangle using the law of sines (wherever possible), with steps shown.

Related calculator: Law of Cosines Calculator

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Solve the triangle, if $b = 3$, $A = 60^0$, $B = 45^0$.

### Solution

According to the law of sines: $\frac{a}{\sin{\left(A \right)}} = \frac{b}{\sin{\left(B \right)}}$.

In our case, $\frac{a}{\sin{\left(60^0 \right)}} = \frac{3}{\sin{\left(45^0 \right)}}$.

Thus, $a = \frac{3 \sin{\left(60^0 \right)}}{\sin{\left(45^0 \right)}} = \frac{3 \sqrt{6}}{2}$.

The third angle is $C = 180^0 - \left(A + B\right)$.

In our case, $C = 180^0 - \left(60^0 + 45^0\right) = 75^0$.

According to the law of sines: $\frac{c}{\sin{\left(C \right)}} = \frac{b}{\sin{\left(B \right)}}$.

In our case, $\frac{c}{\sin{\left(75^0 \right)}} = \frac{3}{\sin{\left(45^0 \right)}}$.

Thus, $c = \frac{3 \sin{\left(75^0 \right)}}{\sin{\left(45^0 \right)}} = \frac{3 \left(1 + \sqrt{3}\right)}{2}$.

The area is $S = \frac{1}{2} a b \sin{\left(C \right)} = \left(\frac{1}{2}\right)\cdot \left(\frac{3 \sqrt{6}}{2}\right)\cdot \left(3\right)\cdot \left(\sin{\left(75^0 \right)}\right) = \frac{9 \left(\sqrt{3} + 3\right)}{8}.$

The perimeter is $P = a + b + c = \frac{3 \sqrt{6}}{2} + 3 + \frac{3 \left(1 + \sqrt{3}\right)}{2} = \frac{3 \left(\sqrt{3} + \sqrt{6} + 3\right)}{2}$.

$a = \frac{3 \sqrt{6}}{2}\approx 3.674234614174767$A

$b = 3$A

$c = \frac{3 \left(1 + \sqrt{3}\right)}{2}\approx 4.098076211353316$A

$A = 60^0$A

$B = 45^0$A

$C = 75^0$A

Area: $S = \frac{9 \left(\sqrt{3} + 3\right)}{8}\approx 5.323557158514987$A.

Perimeter: $P = \frac{3 \left(\sqrt{3} + \sqrt{6} + 3\right)}{2}\approx 10.772310825528083$A.