Find the exact value of the expression, if possible. Do not use a calculator.
-$\cos \left( \tan ^ { - 1 } \frac { 1 } { 5 } \right)$
A) $\frac { 5 \sqrt { 26 } } { 26 }$
B) $\frac { \sqrt { 26 } } { 5 }$
C) $\frac { 5 } { 26 }$
D) $\frac { 1 } { 5 }$

Question

Quizplus · Accepted Answer

To find the exact value of $\cos(\tan^{-1}(\frac{1}{5}))$, we can visualize a right triangle where the opposite side over the adjacent side (the definition of tangent) equals $\frac{1}{5}$. If we let the opposite side be 1 and the adjacent side be 5, then by the Pythagorean theorem, the hypotenuse is $\sqrt{1^2 + 5^2} = \sqrt{26}$. The cosine of an angle is the adjacent side over the hypotenuse, so in this case, $\cos(\tan^{-1}(\frac{1}{5})) = \frac{5}{\sqrt{26}}$, which simplifies to $\frac{5\sqrt{26}}{26}$ when rationalized.

Find the Exact Value of the Expression, If Possible cos⁡(tan⁡−115)\cos \left( \tan ^ { - 1 } \frac { 1 } { 5 } \right)cos(tan−151​)

Find the Exact Value of the Expression, If Possible $\cos \left( \tan ^ { - 1 } \frac { 1 } { 5 } \right)$