Debug

Ambiguous case of the Law of sines

Explained with examples

Example of Zero Triangles Possible

Use the Law of Sines to solve for $$m \angle A$$
Triangle 1  Ambiguous case Zero Triangles
Given
$$ BC = 23 \\ AC = 3 \\ \angle B = 44^{\circ}$$.
Work

$ \frac{sin( A)}{ 23 } = \frac{sin(44)}{3} \\ sin( A) = \frac{23 \cdot sin(44)}{3} \\ sin( A) = \red { 5.32571417} \\ \boxed{\text{ No Solution}} $

As you can see $$ \angle A $$ is 'impossible' because the sine of an angle cannot be equal to 5.3. (Remember the greatest value that the sine of an angle can have is 1)

Visual of Zero Triangles Possible

Look what happens when you try to draw a line from B to A at a $$44^{\circ}$$ angle.  Ambiguous case Zero Triangles

There is no way to do it

Example 1

For $$\triangle ABC$$, $$ a = 6, b =10$$, and $$m \angle A = 42^{\circ}$$. How many Triangles can be formed?

Work

$$ \frac{sin ( B )}{b} = \frac{sin ( A )}{a} \\ \frac{sin (\red B)}{10} = \frac{sin (42^{\circ})}{6} \\ sin(\red B)= \frac{10 \cdot sin (42^{\circ})}{6} \\ sin(\red B)= 1.11522 \\ \boxed{\text{No Solution}} $$

Answer:

Zero Triangles . The maximum value of the sine function is 1.

$$ sin(B)= \cancel {\red {1.11522}} $$

(from our free downloadable worksheet )

Practice Problem

For $$\triangle DEF$$, $$ e=27,f=12$$, and $$m \angle F = 37^{\circ}$$. Find all possible $$m\angle E$$ to the nearest degree.

Work

$$ \frac{sin ( E )}{e} = \frac{sin ( F )}{f} \\ \frac{sin (\red E)}{27} = \frac{sin (37^{\circ})}{12} \\ sin(\red E)= \frac{27 \cdot sin (37^{\circ})}{12} \\ sin(\red B)= 1.3541 \\ \boxed{\text{No Solution}} $$

Answer:

There is no possible $$\angle E $$ with the given dimensions . The maximum value of the sine function is 1.

$$ sin(B)= \cancel {\red {1.3541}} $$

(from our free downloadable worksheet )

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