Expert: Paul Klarreich - 3/3/2007

Question
How do you find a ratio for a cosine of 30degrees?

Answer
Questioner:   nicole
Category:  Advanced Math
 
Subject:  trig
Question:  How do you find a ratio for a cosine of 30 degrees?
...........................................
Hi, Nicole,

You could just push a couple of buttons on your calculator and .. Oh, I forgot -- nobody buys those things any more .. I mean on your Windows calculator.

But I think your teacher wants more than that.  [So would I, if you were my student.]

Here's what you do: (I can't draw diagrams on this clumsy interface, so you will have to do it.)

I. Construct equilateral triangle ABD, such that each side is equal to 2 (miles, inches, whatever), and put A at the top, B at the left, D at the right.  Naturally, each angle is 60 degrees.  [I won't write degrees any more.]

II. Draw AC perpendicular to BD.  Use your knowledge of geometry to conclude that C bisects BD, and so BC = 1 unit.

III. You now have two triangles ABC and ADC.  Erase triangle ADC and concentrate on ABC.  Note that angle A is now 30, angle B is 60, angle C is 90.  ABC is a right triangle.

IV. You have AB = 2, the hypotenuse, BC = 1, a leg, and the Pythagorean Theorem lets you compute AC = sqrt(3).

V. Now use this triangle to compute all the trigonometric functions of either 30 or 60, using the two legs and the hypotenuse.

For example:

VIEW THIS IN COURIER FONT.

        adjacent leg
cos 60 = -------------
        hypotenuse

        BC  
cos 60 = ---
        AB

         1
cos 60 = ---
         2
That's it.