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Which of the following statements are correct for the angle 20π3\frac{20\pi}{3} radians? (without calculator)\newlineI. The value of \sec \frac{20\pi}{3}<0\newlineII. The value of \cosec \frac{20\pi}{3}<0\newline(A) Only I\newline(B) Only II\newline(C) Both I and II\newline(D) Neither

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Q. Which of the following statements are correct for the angle 20π3\frac{20\pi}{3} radians? (without calculator)\newlineI. The value of sec⁡20π3<0\sec \frac{20\pi}{3}<0\newlineII. The value of cosec⁡20π3<0\cosec \frac{20\pi}{3}<0\newline(A) Only I\newline(B) Only II\newline(C) Both I and II\newline(D) Neither
  1. Find Equivalent Angle: First, we need to find the equivalent angle for 20π3\frac{20\pi}{3} in the interval from 00 to 2π2\pi because trigonometric functions repeat every 2π2\pi radians. To do this, we divide 20π3\frac{20\pi}{3} by 2π2\pi to find how many full rotations plus an additional angle it contains.\newline20π3÷2π=103÷2=106=53\frac{20\pi}{3} \div 2\pi = \frac{10}{3} \div 2 = \frac{10}{6} = \frac{5}{3}\newlineThis means that 20π3\frac{20\pi}{3} radians is equivalent to 53\frac{5}{3} of a full rotation plus an additional angle.
  2. Calculate Additional Angle: Next, we calculate the additional angle by multiplying the fraction of the rotation (5/3)(5/3) by 2π2\pi.\newlineAdditional angle = (5/3)×2π=10π/3(5/3) \times 2\pi = 10\pi/3\newlineSince 10π/310\pi/3 is greater than 2π2\pi, we subtract 2π2\pi to find the angle within the interval from 00 to 2π2\pi.\newline10π/3−2π=10π/3−6π/3=4π/310\pi/3 - 2\pi = 10\pi/3 - 6\pi/3 = 4\pi/3\newlineSo, the angle 20π/320\pi/3 is coterminal with 2π2\pi00 radians.
  3. Determine Signs of Secant and Cosecant: Now, we need to determine the signs of secant and cosecant for the angle 4π3\frac{4\pi}{3}. The angle 4π3\frac{4\pi}{3} lies in the third quadrant of the unit circle, where both cosine and sine are negative.
  4. Secant and Cosecant Signs: Since secant is the reciprocal of cosine, and cosine is negative in the third quadrant, sec⁡(4π3)\sec(\frac{4\pi}{3}) is also negative. Therefore, \sec(\frac{20\pi}{3}) < 0 is a correct statement.
  5. Final Answer: Similarly, cosecant is the reciprocal of sine, and sine is negative in the third quadrant, so csc⁡(4π3)\csc(\frac{4\pi}{3}) is also negative. Therefore, \csc(\frac{20\pi}{3}) < 0 is a correct statement.
  6. Final Answer: Similarly, cosecant is the reciprocal of sine, and sine is negative in the third quadrant, so csc⁡(4π3)\csc(\frac{4\pi}{3}) is also negative. Therefore, \csc(\frac{20\pi}{3}) < 0 is a correct statement.Both statements I and II are correct, so the correct answer is C, Both I and II.

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