How do you evaluate #sec(sec^-1((2sqrt3)/3))# without a calculator?

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To evaluate ( \sec(\sec^{-1}(\frac{2\sqrt{3}}{3})) ) without a calculator, recognize that ( \sec^{-1}(\frac{2\sqrt{3}}{3}) ) represents an angle whose secant value is ( \frac{2\sqrt{3}}{3} ).

Given that ( \sec(\theta) = \frac{1}{\cos(\theta)} ), and the cosine of an angle represents the ratio of the adjacent side to the hypotenuse in a right triangle, consider a right triangle where the adjacent side is ( 2\sqrt{3} ) and the hypotenuse is 3.

Using the Pythagorean theorem, you can find the opposite side:

[ (\text{opposite side})^2 = (\text{hypotenuse})^2 - (\text{adjacent side})^2 ]

[ (\text{opposite side})^2 = (3)^2 - (2\sqrt{3})^2 ]

[ (\text{opposite side})^2 = 9 - 12 ]

[ (\text{opposite side})^2 = -3 ]

Since the square of the length of a side of a triangle cannot be negative, this means that there's no real solution for the length of the opposite side. Therefore, the angle whose secant value is ( \frac{2\sqrt{3}}{3} ) is not within the domain of the inverse secant function for real numbers. Thus, ( \sec^{-1}(\frac{2\sqrt{3}}{3}) ) is undefined, and consequently, ( \sec(\sec^{-1}(\frac{2\sqrt{3}}{3})) ) is also undefined.

To evaluate sec(sec^-1((2sqrt3)/3)) without a calculator, first note that sec(x) is the reciprocal of cos(x). Therefore, sec(sec^-1((2sqrt3)/3)) is equivalent to 1/cos(sec^-1((2sqrt3)/3)).

Now, recall the definition of the inverse trigonometric functions: sec^-1(x) represents the angle whose secant is x. So, sec^-1((2sqrt3)/3) represents an angle whose secant is ((2sqrt3)/3).

Since secant is the reciprocal of cosine, and we know that cosine is adjacent over hypotenuse in a right triangle, construct a right triangle where the adjacent side is 2 and the hypotenuse is 3, so that the cosine of one of the acute angles in this triangle is ((2sqrt3)/3).

Now, use the Pythagorean theorem to find the opposite side of this triangle. Then, use the definition of cosine to find the value of cosine of the angle whose secant is ((2sqrt3)/3).

Finally, evaluate 1/cosine of this angle to find the value of sec(sec^-1((2sqrt3)/3)).