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Cos x or sin x or tan x

  1. Sep 20, 2004 #1
    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.
  2. jcsd
  3. Sep 20, 2004 #2
    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.
  4. Sep 20, 2004 #3
    Yeah, you just have to memorize that crap. You won't need a calculator
  5. Sep 20, 2004 #4
    Or you can learn the expanded series notation(infinite series) and just start adding terms.
  6. Sep 20, 2004 #5


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    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.
  7. Sep 20, 2004 #6
    Thanks a lot, all I've to do is play around with the equations now, thanks Hal and guys.
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