use a sketch to find the exact value of the following expression.\n\n cos left \tan ^ { - 1 } left( - \frac…

Explanation:

Step1: Let (\theta=\tan^{- 1}\left(-\frac{2}{7}\right))

By the definition of the inverse - tangent function, (\tan\theta=-\frac{2}{7}), and (\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)). Since (\tan\theta<0), (\theta\in\left(-\frac{\pi}{2},0\right)). We know that (\tan\theta = \frac{y}{x}=-\frac{2}{7}) (let (y=-2) and (x = 7)). Then, by the Pythagorean theorem (r=\sqrt{x^{2}+y^{2}}), where (r>0).

Step2: Calculate (r)

(r=\sqrt{x^{2}+y^{2}}=\sqrt{7^{2}+\left(-2\right)^{2}}=\sqrt{49 + 4}=\sqrt{53})

Step3: Calculate (\cos\theta)

By the definition of the cosine function (\cos\theta=\frac{x}{r}). Since (x = 7) and (r=\sqrt{53}), (\cos\left[\tan^{-1}\left(-\frac{2}{7}\right)\right]=\cos\theta=\frac{7}{\sqrt{53}}). Rationalizing the denominator, we get (\frac{7\sqrt{53}}{53})

Answer:

(\frac{7\sqrt{53}}{53})