Question 4
Using the Law of Sines to solve the all possible triangles if $\angle B=50^{\circ}, a=102, b=40$. If no answer exists, enter DNE for all answers.
$\angle A$ is degreeß;
$\angle C$ is degrees;
\[
c=
\]
Assume $\angle A$ is opposite side $a, \angle B$ is opposite side $b$, and $\angle C$ is opposite side $c$
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\(\boxed{\text{DNE}}\)
Step 1 :Use the Law of Sines to find \(\angle A\): \(\frac{a}{\sin A} = \frac{b}{\sin B}\)
Step 2 :Rearrange for \(\sin A\): \(\sin A = \frac{a \sin B}{b}\)
Step 3 :Substitute the given values: \(\sin A = \frac{102 \sin 50^\circ}{40}\)
Step 4 :Calculate \(\sin A\) to get approximately 1.28
Step 5 :Since the sine of an angle cannot be greater than 1, there are no possible triangles with the given measurements
Step 6 :Therefore, \(\angle A\), \(\angle C\), and \(c\) are all DNE (Does Not Exist)
Step 7 :\(\boxed{\text{DNE}}\)