Law of Sines

The Law of Sines tells us that the ratio between the sine of an angle and the side opposite will be constant for any angle in a triangle.

sinAa=sinBb=sinCc\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}

If we have at least two sides and an angle or two angles and a side, we can use it to find the missing values.



Consider a triangle with angles AA, BB and CC and sides aa, bb and cc, where aa is the side opposite AA, bb opposite BB and cc opposite CC.

If C=42°C = 42°, c=15cmc = 15cm and b=1cmb = 1cm, solve the triangle.


We can start by finding BB using the Law of Sines:

Since we know that sin42°15cm=sinB1cm\frac{\sin 42°}{15cm} = \frac{\sin B}{1cm}

We can calculate sin42°15=0.0446\frac{\sin 42°}{15} = 0.0446

sinB1=0.0446\frac{\sin B}{1} = 0.0446

Since dividing by 1 equals the numerator, we know:

sinB=0.0446\sin B = 0.0446

Then use arcsin\arcsin to find BB

B=arcsin(0.0446)B = arcsin(0.0446)

B=2.557°B = 2.557°

We can now find AA since we know that all the angles in a triangle add up to 180°180°

A=180°42°2.56°A = 180° - 42° - 2.56°

A=135.44°A = 135.44°

Now to find aa, we can use the Law of Sines again:

sin42°15cm=sin(135.44)a\frac{\sin 42°}{15cm} = \frac{\sin(135.44)}{a}

Calculate the known values:

0.0446=0.702/a0.0446 = 0.702 / a

Multiply both sides by A:

a×0.0446=0.702a \times 0.0446 = 0.702

a=0.7020.0446=15.74cma = \frac{0.702}{0.0446} = 15.74cm

Now that we have the missing values, we can use the Law of Sines to check that all ratios are equal:

sin(135.44°)15.74=0.045\frac{sin(135.44°)}{15.74} = 0.045

sin(2.56°)1=0.045\frac{sin(2.56°)}{1} = 0.045

sin(42°)15=0.045\frac{sin(42°)}{15} = 0.045


A=135.44°A = 135.44°

B=2.56°B = 2.56°

C=42°C = 42°

a=15.74cma = 15.74cm

b=1cmb = 1cm

c=15cmc = 15cm

See also Law Of Cosines.