Question

Question #f6d94

Answer 1

`cos(b)=sqrt(289/338) ~~0.924680985 ("radians")`

Explanation:

Here are 2 ways you could do this.

Method 1: ...almost feels like cheating ;)
If `tan(2b)=119/120`
use your calculator (or a spreadsheet) to evaluate:
`color(white)("XXX")2b=arctan(119/120)~~0.7812140874`
which implies (after dividing by 2)
`color(white)("XXX")b=0.3906070437`
Then
use your calculator again to find
`color(white)("XXX")cos(b)=cos(0.3906070437)~~0.9246780985`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Method 2: possibly more virtuous (???)
If `tan(2b)=119/120`
then we can think of angle `(2b)` as being the angle of a right triangle in standard position with the horizontal (`x`) component equal to `120` and the vertical (`y`) component equal to `119`.
The hypotenuse would be equal to `sqrt(120^2+119^2)=169`
(and, yes, I did use a calculator to discover this).

`cos(2b)=("horizontal")/("hypotenuse")=120/169`

Then, if we remember the double angle formula:
`color(white)("XXX")cos(2b)=2cos^2(b)-1`
we can re-arrange the terms to get
`color(white)("XXX")cos(b)=sqrt((cos(2b)+1)/2)`

`color(white)("XXXXXX")=sqrt((120/169+1)/2)`

`color(white)("XXXXXX")=sqrt((289/169)/2)`

`color(white)("XXXXXX")=sqrt(289/338)`

and with the aid of my calculator (again)
`color(white)("XXXXXX")~~0.924680985`