Final answer:
To find the exact value of tan 20 in simplest radical form, use the trigonometric identity tan x = sin x / cos x. Substitute the given value of cos 0 to find the value of sin 0. Then, substitute the values into the equation tan 20 = sin 20 / cos 0 and simplify to get the exact value of tan 20.
Step-by-step explanation:
To find the exact value of tan 20 in simplest radical form, we can use the trigonometric identity tan x = sin x / cos x. Since the given information is about cos 0, we need to find the value of sin 0 first. In Quadrant I, sin x is positive, so we can express sin 0 as √(1 - cos^2 0). Substituting the given value of cos 0 into the equation, we get sin 0 = √(1 - (4/√27)^2).
Next, we can find the value of tan 20 by substituting the values of sin 0 and cos 0 into the equation: tan 20 = sin 20 / cos 0. Simplifying the expression further, we get tan 20 = (√(1 - (4/√27)^2)) / (4/√27).
By rationalizing the denominator and simplifying, the exact value of tan 20 in simplest radical form is (√27 - 4√3) / 4.