Solve for x: arctan(x) = arccos(5/13) Solve for [0, 2π)... How?

Question

3482 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-02-26T09:59:22

$arctanx=arccos(5/13)=67.38^@$ or $292.62^@$ and $x=+-12/5=+-2.4$

$arctanx$ is the angle whose tangent ratio is $x$ .

$arccos(5/13)$ is the angle whose cosine ratio is $5/13$ .

As we have $arctanx=arccos(5/13)$ ,

this means the two angles whose tangent ratio is $x$ and the other whose cosine ratio is $5/13$ are identical.

This means $x$ is the tangent of the angle, whose cosine ratio is $5/13$ . Let the angle be $A$ .

As cosine ratio is $5/13$ , we have $cosA=5/13=$ and hence using a scientific calculator, we get $A=67.38^@$ or $360^@-67.38^@=292.62^@$

i.e. $arctanx=arccos(5/13)=67.38^@$ or $292.62^@$

$secA=13/5$ and $tanA=sqrt(sec^2A-1)=sqrt((13/5)^2-1)$

= $+-12/5$ i.e. $x=+-12/5$

Note that while $tan67.38^@=12/5$ , $tan292.62^@=12/5$