SOLUTION: Use an addition or subtraction formula to find the exact value of the expression. Tan(-5pie/12) The formula is tan(s-t)=tan s + tan t/ 1+tans tant or tan(s+t)= tan s-tan t

Algebra ->  Trigonometry-basics -> SOLUTION: Use an addition or subtraction formula to find the exact value of the expression. Tan(-5pie/12) The formula is tan(s-t)=tan s + tan t/ 1+tans tant or tan(s+t)= tan s-tan t      Log On


   

Question 277603: Use an addition or subtraction formula to find the exact value of the expression.
Tan(-5pie/12)
The formula is tan(s-t)=tan s + tan t/ 1+tans tant
or
tan(s+t)= tan s-tan t/ 1+tans tant
Answer by nabla(475) About Me  (Show Source):
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Try:
To make this more natural, remember that the primitive tangent function has period of pi. So we have Tan(-5pi/12) congruent to Tan(7pi/12) mod pi.
Now: What functions do we easily know the value to for trigonometrics? pi/3 is certainly one. So, pi/3=4pi/12---> We need to know 3pi/12=pi/4 ! This is also an easy find.
So, we take Tan(7pi/12)=Tan(3pi/12 + 4pi/12) Now we use your formulae:
Tan(3pi/12 + 4pi/12)=[Tan(pi/4)-Tan(pi/3)]/(1+Tan(pi/4)Tan(pi/3))
Now it remains to remember what Tan(pi/4) and Tan(pi/3) are. We use the definition of Tangent.
Tan(x)=Sin(x)/Cos(x).
So Tan(pi/4)=1.
Tan(pi/3)=(sqrt(3)/2)/(1/2)=sqrt(3)
Finally, substitute into our expression:
(1-sqrt(3))/(1+sqrt(3))
Now, simplify:
%281-sqrt%283%29%29%5E2%2F%281-3%29=%281-2%2Asqrt%283%29%2B3%29%2A%281%2F%28-2%29%29=-2-sqrt%283%29
This could be done quicker, but I hope you can see the full details of what is going on through this procedure.