Multiple choice

If x is an angle in quadrant III and tan (x - 30°) = cot x, find x.

  1. 240°

  2. 225°

  3. 210°

  4. 60°
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A Correct answer
Explanation

tan(x - 30) = cot(x) = tan(90 - x). Thus, x - 30 = 90 - x + 180k. 2x = 120 + 180k. For k=1, 2x = 300, x = 150. For k=2, 2x = 480, x = 240. Since x is in quadrant III (180 to 270), 240 is the correct solution.

AI explanation

Using the identity that cot x equals tan(90 minus x), the equation becomes tan(x minus 30) equals tan(90 minus x). The general solution is x minus 30 equals 90 minus x plus 180n, which simplifies to 2x equals 120 plus 180n, or x equals 60 plus 90n. For the angle to be in the third quadrant between 180 and 270 degrees, set n equal to 2, giving x equals 240 degrees.

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