Law of Sines - Some Applications

Some Applications

• The sine law can be used to prove the angle sum identity for sine when α and β are each between 0 and 90 degrees.

To prove this, make an arbitrary triangle with sides a, b, and c with corresponding arbitrary angles A, B and C. Draw a perpendicular to c from angle C. This will split the angle C into two different angles, α and β, that are less than 90 degrees, where we choose to have α to be on the same side as A and β be on the same side as B. Use the sine law identity that relates side c and side a. Solve this equation for the sine of C. Notice that the perpendicular makes two right angles triangles, also note that sin(A) = cos(α), sin(B) = cos(β) and that c = a sin(β) + b sin(α). After making these substitutions you should have sin(C) =sin(α + β) = sin(β)cos(α) + (b/a)sin(α)cos(α). Now apply the sine law identity that relates sides b and a and make the substitutions noted before. Now substitute this expression for (b/a) into the original equation for sin(α + β) and you will have the angle sum identity for α and β in terms of sine.

The only thing that was used in the proof that was not a definition was the sine law. Thus the sine law is equivalent to the angle sum identity when the angles sum is between 0 and 180 degrees and when each individual angle is between 0 and 90 degrees.

• The sine law along with the prosthaphaeresis and shift identities can be used to prove the law of tangents and Mollweide's formulas (Dresden 2009, Plane Trigonometry pg. 76–78 ).