Find the exact value of sec ((7pi)/(4)) in simplest form with a rational denominator.

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Find the exact value of  sec ((7pi)/(4)) in simplest form with a rational denominator.

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Understand Secant Definition: To find the exact value of sec((7pi)/4), we need to understand that secant is the reciprocal of cosine. So, we first find the cosine of (7pi)/4 and then take its reciprocal.Identify Quadrant of Angle: The angle (7pi)/4 is an angle that lies in the fourth quadrant of the unit circle, where cosine values are positive. Since the unit circle is periodic with a period of 2pi, we can subtract 2pi from (7pi)/4 to find a coterminal angle that is easier to evaluate. (7pi)/4 - 2pi = (7pi - 8pi)/4 = (-pi)/4Calculate Cosine Value: The cosine of (-pi)/4 is the same as the cosine of pi/4 because cosine is an even function, which means cos(-x) = cos(x). The cosine of pi/4 is well-known and equals sqrt(2)/2.Find Reciprocal for Secant: Now, we take the reciprocal of cos((7pi)/4), which is the same as the reciprocal of cos(pi/4), to find sec((7pi)/4). sec((7pi)/4) = 1 / (sqrt(2)/2)Simplify and Finalize: To simplify the expression and eliminate the radical from the denominator, we multiply the numerator and denominator by sqrt(2). sec((7pi)/4) = (1 * sqrt(2)) / (sqrt(2)/2 * sqrt(2)) = sqrt(2) / (2/2) = sqrt(2)

Find the exact value of $\sec \frac{7 \pi}{4}$ in simplest form with a rational denominator.

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Find the exact value of $\sec \frac{7 \pi}{4}$ in simplest form with a rational denominator.