Answer:
[tex] \tan(\cos^{-1}(-5/7)) = -\dfrac{2 \sqrt{6}}{5} \textsf{ or } 0.9797958971 [/tex]
Step-by-step explanation:
To find [tex] \tan(\cos^{-1}(-\frac{5}{7})) [/tex], we first need to determine the angle whose cosine is [tex] -\dfrac{5}{7} [/tex].
Let [tex] \theta = \cos^{-1}(-\frac{5}{7}) [/tex].
This means [tex] \cos(\theta) = -\dfrac{5}{7} [/tex].
Now, we can use the Pythagorean identity for trigonometric functions:
[tex] \sin^2(\theta) + \cos^2(\theta) = 1 [/tex]
Given that [tex] \cos(\theta) = -\dfrac{5}{7} [/tex], we can solve for [tex] \sin(\theta) [/tex]:
[tex] \sin^2(\theta) + \left(-\dfrac{5}{7}\right)^2 = 1 [/tex]
[tex] \sin^2(\theta) + \dfrac{25}{49} = 1 [/tex]
[tex] \sin^2(\theta) = 1 - \dfrac{25}{49} [/tex]
[tex] \sin^2(\theta) = \dfrac{49}{49} - \dfrac{25}{49} [/tex]
[tex] \sin^2(\theta) = \dfrac{24}{49} [/tex]
[tex] \sin(\theta) = \pm \sqrt{\dfrac{24}{49}} [/tex]
Since [tex] \cos(\theta) = -\dfrac{5}{7} [/tex] (negative), the corresponding [tex] \sin(\theta) [/tex] should be positive (due to the negative sign in the second quadrant where cosine is negative).
[tex] \sin(\theta) = \sqrt{\dfrac{24}{49}} = \dfrac{2 \sqrt{6}}{7} [/tex]
Now, we can calculate [tex] \tan(\theta) [/tex]:
[tex] \tan(\theta) = \dfrac{\sin(\theta)}{\cos(\theta)} = \dfrac{\dfrac{2 \sqrt{6}}{7}}{-\dfrac{5}{7}} = -\dfrac{2 \sqrt{6}}{5} [/tex]
Therefore,
[tex] \tan(\cos^{-1}(-5/7)) = -\dfrac{2 \sqrt{6}}{5} \textsf{ or } 0.9797958971 [/tex]
