3 Rules necessary for an Ambiguous Sine Rule Case

Ambiguous case of the sine rule is when there are two possible triangles that satisfies the given information.

For example,

sin R / r = sin Q / q
Put in the values we know:sin R / 41 = sin(39º)/28
Multiply both sides by 41:sin R = (sin39º/28) × 41
Calculate:sin R = 0.9215...
Inverse Sine:R = sin-1(0.9215...)
R = 67.1º




However, sin (112.9) = sin 67.1. As such, you can have the two triangles shown below.



The three conditions required for the ambiguous case to happen (before this we need to have two sides e.g. b & a and the non-included angle)

 a < c  
 a < perpendicular ht. of point C to c
given angle (the non-included one) is acute







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