If sec(θ)=54 and 0°<θ<90°, what is tan(θ)?Write your answer in simplified, rationalized form.tan(θ)

The secant of an angle θ, denoted as sec(θ), is defined as the reciprocal of the cosine of θ. So, if sec(θ) = 54, then cos(θ) = 1/54.

The tangent of an angle θ, denoted as tan(θ), is defined as the sine of θ divided by the cosine of θ. But, in a right triangle, the sine of an angle is the length of the opposite side divided by the length of the hypotenuse, and the cosine of an angle is the length of the adjacent side divided by the length of the hypotenuse.

So, if we let the length of the hypotenuse be 54 (since cos(θ) = 1/54), and the length of the adjacent side be 1 (since the cosine of an angle is the length of the adjacent side divided by the length of the hypotenuse), then the length of the opposite side (which is the sine of θ) can be found using the Pythagorean theorem: (hypotenuse)^2 = (adjacent side)^2 + (opposite side)^2.

Substituting the given lengths, we get (54)^2 = (1)^2 + (opposite side)^2. Solving for the opposite side, we get opposite side = sqrt[(54)^2 - (1)^2] = sqrt[2915] = 54sqrt[1 - (1/54)^2].

Therefore, tan(θ) = sin(θ)/cos(θ) = (54sqrt[1 - (1/54)^2])/1 = 54sqrt[1 - (1/54)^2].