Problem 24 - Calculus of a Single Variable

Question

Evaluate each expression without using a calculator. (Hint: See Example 3.) (a) \(\sec \left[\arctan \left(-\frac{3}{5}\right)\right]\) (b) \(\tan \left[\arcsin \left(-\frac{5}{6}\right)\right]\)

Step-by-Step Solution

Answer

The answers are (a) \(\frac{4}{5}\) and (b) \(\frac{-5}{\sqrt{11}}\).

Step 1: Evaluate (a)

The secant function is the reciprocal of the cosine function, so if we can find \(\cos(\arctan(-3/5))\), we can take the reciprocal to find \(\sec(\arctan(-3/5))\). We know from trigonometry that \(\cos(\arctan(x)) = \frac{1}{\sqrt{1 + x^2}}\). Therefore, \(\cos(\arctan(-3/5)) = \frac{1}{\sqrt{1 + (-3/5)^2}}\).

Step 2: Simplify (a)

Calculating \(\cos(\arctan(-3/5)) = \frac{1}{\sqrt{1 + (-3/5)^2}} = \frac{1}{\sqrt{1 + 9/25}} = \frac{1}{\sqrt{16/25}} = \frac{1}{4/5} = \frac{5}{4}\). Therefore, \(\sec(\arctan(-3/5)) = \frac{1}{\cos(\arctan(-3/5))} = \frac{1}{5/4} = \frac{4}{5}\).

Step 3: Evaluate (b)

To find \(\tan(\arcsin(-5/6))\), we use the identity \(\tan(\arcsin(x)) = \frac{x}{\sqrt{1 - x^2}}\). Therefore, \(\tan(\arcsin(-5/6)) = \frac{-5/6}{\sqrt{1 - (-5/6)^2}}\).

Step 4: Simplify (b)

Calculating \(\tan(\arcsin(-5/6)) = \frac{-5/6}{\sqrt{1 - (-5/6)^2}} = \frac{-5/6}{\sqrt{1 - 25/36}} = \frac{-5/6}{\sqrt{11/36}} = \frac{-5/6}{\sqrt{11}/6} = \frac{-5}{\sqrt{11}}\).