Law of Sines Calculator

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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Your Input

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, $$$c = \frac{3 \sin{\left(75^0 \right)}}{\sin{\left(45^0 \right)}} = 3 \sqrt{2} \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)$$$.

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 $$$P = a + b + c = \frac{3 \sqrt{6}}{2} + 3 + 3 \sqrt{2} \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right) = \frac{3 \left(\sqrt{3} + \sqrt{6} + 3\right)}{2}.$$$

Answer

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

$$$b = 3$$$A

$$$c = 3 \sqrt{2} \left(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)\approx 4.098076211353316$$$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.