In triangle ABC, angle BCA = pi/6, AB = \(16\), and AC = \(15\)
Find the sum of all distinct possible values of BC. If there are no possible values, enter 0.
+15001 Find the sum of all distinct possible values of BC.
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\(\gamma=\frac{\pi}{6}=30^o\)
law of sine
\(\frac{sin\ \gamma}{16}=\frac{sin\ \beta}{15}\\ \frac{sin\ 30^o}{16}=\frac{sin\ \beta}{15}\\ sin\ \beta=\frac{15\cdot sin\ 30^o}{16}\\ \beta=arcsin\ \frac{15\cdot 0.5}{16}\\ \color{blue}\beta \in\{27.953^o,62.047^o\}\)
\(\alpha = 180^o-30^o -\{62.047^o,27.953^o\} \\ \alpha \in \{87.9532^o,122.0468^o\}\)
\(\frac{a_1}{sin\ \alpha_1}=\frac{16}{sin\ 30^o}\\ a_1=\frac{16\cdot sin\ 87.9532^o}{0.5}\\ \color{blue}a_1=31.9796\)
\(\frac{a_2}{sin\ \alpha_2}=\frac{16}{sin\ 30^o}\\ a_2=\frac{16\cdot sin\ 122.0468^o}{0.5}\\ \color{blue}a_2=27.1237\)
\(The\ sum\ of\ all\ distinct\ possible\ values\ of\ BC\ is\ \color{blue}a_1+a_2=59.1 \)
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