In △ABC, m∠A=23°, a=10, and b=13. Find c to the nearest tenth.

9514 1404 393

  c ∈ {3.4, 20.6}

Step-by-step 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.

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step-by-step explanation: