Question

How do you calculate `Cos^-1(5/13) - Cos^-1(8/17)`?

Answer 1

With a calculator. A TI-8# can do this with 2nd+COS. 13 and 17 are prime numbers, so 5/13 and 8/17 will give you irrational values. This isn't easily doable enough by hand to be something you need to do by hand.

`arccos(5/13) - arccos(8/17) = 67.38^o - 61.93^o = 5.45^o ~~ 0.095 "rad"`

Answer 2

`=cos^(-1)(220/221)`

Explanation:

Another way would be to let `A=cos^(-1)(5/13)` and `B=cos^(-1)(8/17)`

`=>cosA=5/13`

And , `cosB=8/17`

From the identity `cos^2x+sin^2x=1`
We can obtain that,

`sinx=sqrt(1-cos^2x)`

`=>sinA=sqrt(1-(5/13)^2)=sqrt(144/169)=12/13`

Similarly , `=>sinB=sqrt(1-(8/17)^2)=sqrt(225/289)=15/17`

Now, say `" "C=A-B`

`=>cosC=cos(A-B)=cosAcosB+sinAsinB=5/13*8/17+12/13*15/17=40/221+180/221=220/221`

`=>cosC=220/221`

`=>C=cos^(-1)(220/221)`

`=>A-B=cos^(-1)(220/221)`

`=cos^(-1)(5/13)-cos^(-1)(8/17)=cos^(-1)(220/221)`