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If all the angles used in the following statements are in radians, which of the statements are true? \newlineI. \cos(\frac{8\pi}{5}) > \sin(\frac{3\pi}{4}) \newlineII. \cos(\frac{8\pi}{3}) > \tan(\frac{2\pi}{5}) \newline(A) Only I\newline(B) Only II\newline(C) Both I and II\newline(D) Neither

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Q. If all the angles used in the following statements are in radians, which of the statements are true? \newlineI. cos⁡(8π5)>sin⁡(3π4) \cos(\frac{8\pi}{5}) > \sin(\frac{3\pi}{4}) \newlineII. cos⁡(8π3)>tan⁡(2π5) \cos(\frac{8\pi}{3}) > \tan(\frac{2\pi}{5}) \newline(A) Only I\newline(B) Only II\newline(C) Both I and II\newline(D) Neither
  1. Evaluate Statement I: To solve this problem, we will evaluate each statement separately to determine its truth value. We will start with statement I: \cos(8\pi/5)>\sin(3\pi/4). First, we need to find the exact values of cos⁡(8π/5)\cos(8\pi/5) and sin⁡(3π/4)\sin(3\pi/4). The angle 8π/58\pi/5 is in the third quadrant where cosine is negative, and the reference angle is π−8π/5=5π/5−8π/5=−3π/5\pi - 8\pi/5 = 5\pi/5 - 8\pi/5 = -3\pi/5. The cosine of the reference angle is the same as the cosine of the original angle in magnitude but opposite in sign. The angle 3π/43\pi/4 is in the second quadrant where sine is positive, and the reference angle is π−3π/4=4π/4−3π/4=π/4\pi - 3\pi/4 = 4\pi/4 - 3\pi/4 = \pi/4. The sine of the reference angle is the same as the sine of the original angle.
  2. Find Exact Values: Now, let's find the exact values. The cosine of the reference angle −3π/5-3\pi/5 is not one of the standard angles we know the exact value for, so we cannot find an exact value for cos⁡(8π/5)\cos(8\pi/5) without a calculator. However, we know that in the third quadrant, cosine is negative, so cos⁡(8π/5)\cos(8\pi/5) is negative.\newlineFor sin⁡(3π/4)\sin(3\pi/4), we know that sin⁡(π/4)=2/2\sin(\pi/4) = \sqrt{2}/2, and since 3π/43\pi/4 is in the second quadrant where sine is positive, sin⁡(3π/4)=2/2\sin(3\pi/4) = \sqrt{2}/2.
  3. Compare Values: Comparing the two values, we have a negative value for cos⁡(8π5)\cos(\frac{8\pi}{5}) and a positive value for sin⁡(3π4)\sin(\frac{3\pi}{4}). Therefore, it is clear that cos⁡(8π5)\cos(\frac{8\pi}{5}) is not greater than sin⁡(3π4)\sin(\frac{3\pi}{4}), and statement I is false.
  4. Evaluate Statement II: Next, we will evaluate statement II: \cos(8\pi/3)>\tan(2\pi/5). First, we need to find the exact values of cos⁡(8π/3)\cos(8\pi/3) and tan⁡(2π/5)\tan(2\pi/5). The angle 8π/38\pi/3 is more than 2π2\pi, so we need to subtract 2π2\pi to find the coterminal angle in the range [0,2π)[0, 2\pi). 8π/3−2π=8π/3−6π/3=2π/38\pi/3 - 2\pi = 8\pi/3 - 6\pi/3 = 2\pi/3. The cosine of 2π/32\pi/3 is in the second quadrant where cosine is negative. The angle 2π/52\pi/5 is in the first quadrant where all trigonometric functions are positive, and tan⁡(2π/5)\tan(2\pi/5) is not one of the standard angles we know the exact value for, but we know it will be positive.
  5. Find Exact Values: Since cos⁡(8π3)\cos(\frac{8\pi}{3}) is negative (because it is the same as cos⁡(2π3)\cos(\frac{2\pi}{3}) which is in the second quadrant) and tan⁡(2π5)\tan(\frac{2\pi}{5}) is positive, it is clear that cos⁡(8π3)\cos(\frac{8\pi}{3}) is not greater than tan⁡(2π5)\tan(\frac{2\pi}{5}), and statement II is false.
  6. Compare Values: Both statements I and II are false, so the correct answer is D: Neither.

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