Solve Trigonometry Equation: Tan 285° + Cos 75° + Cot 60°

[markm]

the question is: find the exact value of tan 285 deg + cos 75 deg + cot 60 deg

i tried converting them to radians and got

tan 5pi / 12 + cos 5 pi / 12 + cot pi / 3

as far as i know, only the cot part has some special identity. am i missing something important? we have a long test tomorrow..

thanks in advance for any help. ^_^

 

[markm]

oh, and by exact value the question means no decimal places.

 

[dextercioby]

RULE NUMBER ONE ON THIS FORUM:

DON'T DOUBLE POST!

Daniel.

 

[HallsofIvy]

I didn't see a double post so I won't comment on that.

The denominator in "5pi/12" is 12= 3*4. Hmm, pi/3 and pi/4 tend to be pretty easy!

Can we "analyze" 5pi/12 as a combination of pi/3 and pi/4? That is, can we find
m and n so that 5pi/12= mpi/4+ npi/3? If we multiply that equation by 12/pi, we get
5= 3m+ 4n. Now I think it's just a matter of "trial and error" (actually, there are methods of solving such "Diaphontine equations" but trial and error works here).Trying a few possibilities shows that m= -5, n= 5 gives 3(-5)+ 4(5)= -15+ 20= 5.

Excellent: 5pi/12= 20pi/12- 15pi/12= 5(pi/3)- 5(pi/4) so tan(5pi/12)= tan(5(pi/3)- 5(pi/4) and cos(5pi/12)= cos(5(pi/3)- 5(pi/4)).

Now, do you know any formulas for reducing tan(a-b) and cos(a-b)?

 

[dextercioby]

The same problem,the same "author" and the same me giving the same hint in the HS Homework section...

Daniel.