Can the Numerical Value of Sin or Cos be Derived Without a Calculator?

[haribol]

Assume you are given tan (pi/6). This means sin (pi/6)/cos (pi/6). But my question is, if you are given sin (pi/6) or cos (pi/6), is it possible to derive the numerical value without using calculator?

-Thank you in advance.

 

[Sirus]

You can derive these based on a unit circle triangle (don't remember exactly), but in the end you just have to memorize them anyways for use in trig.

 

[swingkids]

Yeah, you just have to memorize that crap. You won't need a calculator

 

[K.J.Healey]

Or you can learn the expanded series notation(infinite series) and just start adding terms.

 

[HallsofIvy]

Better: Construct an equilateral triangle,with each side of length 2, then drop a perpendicular from the top to the base. That line bisects the angle and the opposite side. Since all three angles of an equilateral triangle are pi/3 radians (or 60 degrees), half of it is pi/6 (or 30 degrees) , while half the base has length 1, we have a right triangle with side opposite the pi/6 angle of length 1, hypotenuse of length 2 and side "near" the pi/6 angle of length (use the Pythagorean theorem) √(4- 1)= √(3).

sin(pi/6)= opposite/hypotenuse= 1/2.
cos(pi/6)= near/hypotenuse= √(3)/2.

tan(pi/6)= opposite/near= 1/√(3)= √(3)/3.
cotan(pi/6)= near/opposite= √(3)/1= √(3).

sec(pi/6)= hypotenuse/near= 2/√(3)= 2√(3)/3.
csc(pi/6)= hypotenuse/opposite= 2/1=2.

Of course, you can also use that to find the trig functions for pi/3, the other angle in the right triangle.

 

[haribol]

Thanks a lot, all I've to do is play around with the equations now, thanks Hal and guys.