10.06 The ambiguous case

As we just saw, the sine rule applies to all triangles, like this one.

We can write it as follows: \dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}

This rule relates each side in a triangle with the sine ratio of the angle opposite to it. For example, if we know the side lengths a and b, and know the size of angle \angle A, we can take the first equality and rearrange it to obtain\sin B = \dfrac{b \sin A}{a},

which lets us determine the sine of B. To find B itself, we need to take the inverse:B = \sin^{-1} \left( \dfrac{b \sin A}{a}\right).

We can then substitute the values of a, b, and A to find a value for B. But in this case, there is more to consider. To see when and why, we are going to dive a little deeper into the geometry of the situation.

Say we have a non-right angled triange where we know the length of two sides and one angle opposite one of the known sides. When the length of the side opposite the known angle is less than the length of the other known side, there are two possible triangles, and two possible values for the other angle.

This is called the ambiguous case.

We can also see this algebraically by looking at the equation we used before:B = \sin^{-1} \left( \dfrac{b \sin A}{a}\right).

This equation can have more than one valid solution (though your calculator will only ever give you one). Luckily for us, the two values of B that are produced always add to 180 \degree. In summary:

If you are trying to find an angle using the sine rule (with two known sides and a known angle), use the sine rule to find one value of B. If the side opposite the known angle is the shorter side, you are in the ambiguous case. Subtract the first value of B that you found from 180 \degree to find the second solution.

Examples

Example 1

Find the value of x in degrees, given that x is obtuse. Round your answer to two decimal places.

Worked Solution

Create a strategy

Use the sine rule: \dfrac{\sin A}{a}=\dfrac{\sin B}{b}.

Apply the idea

\displaystyle \dfrac{\sin x}{18.8}	\displaystyle =	\displaystyle \dfrac{\sin 25 \degree }{12.6}	Substitute A=x, \,a=18.8, \, b=12.6, \, B=25
\displaystyle \sin x	\displaystyle =	\displaystyle 18.8 \times\frac{\sin 25\degree }{12.6}	Multiply both sides by 18.8
\displaystyle \sin x	\displaystyle =	\displaystyle 0.630\,573	Evaluate
\displaystyle x	\displaystyle =	\displaystyle \sin ^{-1}\left(0.630\,573\right)	Take the inverse sine of both sides
	\displaystyle =	\displaystyle 39.09\degree	Evaluate

To find the obtuse angle x we subtract the acute angle from 180\degree:

\displaystyle x	\displaystyle =	\displaystyle 180 - 39.09	Subtract 39.09 from 180
	\displaystyle =	\displaystyle 140.91\degree	Evaluate

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Example 2

\triangle ABC consists of angles A, \, B, and C which appear opposite sides a, \, b, and c respectively where \angle CAB = 53\degree, \, a=12, and b=7.

Which is the most appropriate option?

A

There are two such triangles, one obtuse and one acute.

B

No such triangle exists.

C

Only one triangle exists.

Worked Solution

Create a strategy

Remember that if the side opposite the known angle is the shorter side, then we are in the ambiguous case.

Apply the idea

In \triangle ABC the opposite of the known angle \angle CAB = 53\degree is the side with length a=12.

This is greater than the length of the other known side, b=7. So there is only one possible triangle, and only one possible value for the other angle.

The most appropriate option is option C.

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