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

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Precalculus

Inverses of trigonometric functions

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Q. If all the angles used in the following statements are in radians, which of the statements are true?

\newline

\cos(\frac{8\pi}{5}) > \sin(\frac{3\pi}{4})

\newline

II.

\cos(\frac{8\pi}{3}) > \tan(\frac{2\pi}{5})

\newline

(A) Only I

\newline

(B) Only II

\newline

\newline

(D) Neither

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 $\cos(8\pi/5)$ and $\sin(3\pi/4)$ . The angle $8\pi/5$ is in the third quadrant where cosine is negative, and the reference angle is $\pi - 8\pi/5 = 5\pi/5 - 8\pi/5 = -3\pi/5$ . The cosine of the reference angle is the same as the cosine of the original angle in magnitude but opposite in sign. The angle $3\pi/4$ is in the second quadrant where sine is positive, and the reference angle is $\pi - 3\pi/4 = 4\pi/4 - 3\pi/4 = \pi/4$ . The sine of the reference angle is the same as the sine of the original angle.
Find Exact Values: Now, let's find the exact values. The cosine of the reference angle $-3\pi/5$ is not one of the standard angles we know the exact value for, so we cannot find an exact value for $\cos(8\pi/5)$ without a calculator. However, we know that in the third quadrant, cosine is negative, so $\cos(8\pi/5)$ is negative. $\newline$ For $\sin(3\pi/4)$ , we know that $\sin(\pi/4) = \sqrt{2}/2$ , and since $3\pi/4$ is in the second quadrant where sine is positive, $\sin(3\pi/4) = \sqrt{2}/2$ .
Compare Values: Comparing the two values, we have a negative value for $\cos(\frac{8\pi}{5})$ and a positive value for $\sin(\frac{3\pi}{4})$ . Therefore, it is clear that $\cos(\frac{8\pi}{5})$ is not greater than $\sin(\frac{3\pi}{4})$ , and statement I is false.
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 $\cos(8\pi/3)$ and $\tan(2\pi/5)$ . The angle $8\pi/3$ is more than $2\pi$ , so we need to subtract $2\pi$ to find the coterminal angle in the range $[0, 2\pi)$ . $8\pi/3 - 2\pi = 8\pi/3 - 6\pi/3 = 2\pi/3$ . The cosine of $2\pi/3$ is in the second quadrant where cosine is negative. The angle $2\pi/5$ is in the first quadrant where all trigonometric functions are positive, and $\tan(2\pi/5)$ is not one of the standard angles we know the exact value for, but we know it will be positive.
Find Exact Values: Since $\cos(\frac{8\pi}{3})$ is negative (because it is the same as $\cos(\frac{2\pi}{3})$ which is in the second quadrant) and $\tan(\frac{2\pi}{5})$ is positive, it is clear that $\cos(\frac{8\pi}{3})$ is not greater than $\tan(\frac{2\pi}{5})$ , and statement II is false.
Compare Values: Both statements I and II are false, so the correct answer is D: Neither.