Question

find the approximate value of \\( \cos ( \tan ^ { - 1 } ( - 0.62 ) ) \\).
\\( \cos ( \tan ^ { - 1 } ( - 0.62 ) ) = \\)
(round to four decimal places.)

Explanation:

Step1: Let \(\theta=\tan^{- 1}(-0.62)\)

By the definition of inverse tangent function, \(\tan\theta=-0.62\), and \(\theta\in(-\frac{\pi}{2},\frac{\pi}{2})\). We know that \(\tan\theta=\frac{y}{x}=-0.62\). Using the identity \(\cos\theta=\frac{1}{\sqrt{1 + \tan^{2}\theta}}\)

Step2: Substitute \(\tan\theta=-0.62\) into the formula

$$
LATEXBLOCK0
$$

Step3: Calculate the value

\(\sqrt{1.3844}\approx1.1766\), then \(\cos\theta=\frac{1}{1.1766}\approx0.85\)

Another way:

Step1: Calculate \(\theta=\tan^{-1}(-0.62)\)

Using a calculator, \(\theta=\tan^{-1}(-0.62)\approx - 0.5586\) (in radians)

Step2: Calculate \(\cos\theta\)

\(\cos(-0.5586)=\cos(0.5586)\approx0.85\) (since \(\cos(-x)=\cos x\))

Answer:

\(0.8500\)