Sine Rule – more on the ambiguous case

Sin A / a    =    Sin B / b    =    Sin C / c

So you can use the Sine rule if you have two angles and one side (AAS):

Sin A / a    =    Sin B / x

Or if you have two sides and one angle (SSA):

Sin A / a    =    x / b

BUT remember that if you are given two sides and one angle of a triangle (SSA), you don’t have enough information to know exactly what that triangle looks like:

Look at the diagram.

Is it

Sin 35 / 6cm    =    Sin C / 10cm?

or

Sin 35 / 6cm    =   Sin D / 10cm? 

This is the ambiguous case (ambiguous means “could-be-either”). To identify the ambiguous case, you need to follow these rules:

• If you have AAS, then everything is fine, go ahead and use the Sine rule.
• If you have SSA and the side opposite the angle you have been given is the longer of the two sides you have been given, then everything is fine, go ahead and use the Sine rule.
• Otherwise, you have an ambiguous case and you need to be careful.

What to do about the ambiguous case?

Remember that Sin(x) = Sin(180-x), for instance:

• Sin(80) = Sin(100)
• Sin(45) = Sin(135)
• Sin(10) = Sin(170)
• Sin(0) = Sin(180)

Your missing angle is either x or 180 – x. But…

And this is important…

When you use inverse sine to find your missing angle, it will always give you x rather than 180 – x.

Technically, it will always give you x such -90 <= x < 90.

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