9514 1404 393
Answer:
c ∈ {3.4, 20.6}
Explanation:
The law of cosines can be used to write a quadratic equation in c.
a^2 = b^2 + c^2 -2bc·cos(A)
10^2 = 13^2 + c^2 -2(13)c ·cos(23°)
c^2 -23.933c +69 = 0 . . . . . subtract 100 to put into standard form.
c = (-(-23.933) ±√((-23.933)^2 -4(1)(69)))/(2(1))
c ≈ 11.967 ± √74.199
c ≈ {3.4, 20.6}
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Additional comment
The given angle is opposite the shorter of the two given sides, so we expect two solutions. We could use the law of sines, but this works just as well and gives the two solutions directly, without having to find angle B.