Question

Evaluate the expression without using a calculator. (Hint: Make a sketch of a right triangle, as illustrated in Example \(7 .)\) (a) \(\tan (\operatorname{arccot} 2)\) (b) \(\cos (\operatorname{arcsec} \sqrt{5})\)

Short answer

The answers to the given expressions are: (a) \(\tan (\operatorname{arccot} 2) = \frac{1}{2}\) and (b) \(\cos(\operatorname{arcsec} \sqrt{5}) = \frac{1}{\sqrt{5}}\)

Step-by-step solution

Evaluating the first expression

For the first expression, we need to evaluate \(\tan (\operatorname{arccot} 2)\). The arccotangent function is the inverse of the cotangent function. Therefore, this function returns an angle whose cotangent is 2. Let’s use \(\Theta\) to denote this angle. We now have our right triangle with the adjacent side 2 and the opposite side 1. Remembering the definitions of tangent and cotangent, we have the following relationships in our right triangle: Cotangent, \(\cot(\Theta) = \frac{Adjacent}{Opposite} = 2\) Tangent, \(\tan(\Theta) = \frac{1}{ \cot(\Theta)}\)

Substituting into the Expression

With these relationships established, we can then substitute them back into the original expression, as follows: \(\tan(\operatorname{arccot} 2) = \tan(\Theta) = \frac{1}{2}\)

Evaluating the second expression

For the second expression, similarly, we want to find \(\cos(\operatorname{arcsec} \sqrt{5})\). The arcsecant function is the inverse of the secant function, hence it returns an angle whose secant is \(\sqrt{5}\). If we again let \(\Theta\) denote this angle, our right triangle now has the hypotenuse \(\sqrt{5}\) and the adjacent side 1. Remembering the definition of cosine, \(\cos(\Theta) = \frac{Adjacent}{Hypotenuse} = \frac{1}{\sqrt{5}}\)

Substituting into the Expression

Substituting these relationships back into the original expression gives: \(\cos(\operatorname{arcsec} \sqrt{5}) = \cos(\Theta) = \frac{1}{\sqrt{5}}\)