Multiple choice

In a triangle two sides are of length 20 and 21 and the sine of angle between them is equal to 0.6 then the third side is equal to

  1. $13\ or\ \sqrt{1513}$
  2. $14\ or\ \sqrt{1315}$
  3. $15\ or\ \sqrt{1531}$
  4. $17\ or\ \sqrt{1531}$
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A Correct answer
Explanation

Area = 1/2 * a * b * sin(C). 1/2 * 20 * 21 * 0.6 = 126. Using Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos(C). sin(C) = 0.6, so cos(C) = +/- 0.8. c^2 = 20^2 + 21^2 - 2*20*21*(+/- 0.8) = 400 + 441 - 840*(+/- 0.8) = 841 - 672 = 169 (c=13) or 841 + 672 = 1513 (c=sqrt(1513)).

AI explanation

Using the cosine rule, the third side x squared equals 20 squared plus 21 squared minus 2 times 20 times 21 times cos A, where cos A can be 0.8 or negative 0.8 because sin A is 0.6. For cos A equals 0.8, x squared equals 400 plus 441 minus 672, giving x equals 13. For cos A equals negative 0.8, x squared equals 400 plus 441 plus 672, which equals 1513, so x equals the square root of 1513. Therefore, the third side is 13 or the square root of 1513.

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