Deriving the Coordinates of the pi/6 Family
Date: 06/09/99 at 20:33:40 From: Kelly Subject: Deriving the coordinates of the pi/6 family on the unit circle I need to know how to derive the coordinates of the pi/6 family on the unit circle. I know that you use a reference triangle and use the definition of sin and cos to get the answer. However, I need to know why this works.
Date: 06/10/99 at 12:35:26 From: Doctor Rick Subject: Re: Deriving the coordinates of the pi/6 family on the unit circle Hi, Kelly. Do you understand why sin(pi/4) = sqrt(2)/2? I'll go over it, because this will lead into the angles you asked about. Consider a 45-45-90 triangle with a hypotenuse of 1. Because 2 angles are equal, it is isosceles; both legs have the same length. If you know one leg (a), you know both legs. Let's use the Pythagorean Theorem: /| / | 1 / | / | a / | / | /____________| a 1 = sqrt(a^2 + a^2) = sqrt(2*a^2) = sqrt(2)*a Solve for a: 1/sqrt(2) = a Rationalize the denominator by multiplying numerator and denominator by sqrt(2): sqrt(2)/2 = a So these are the proportions of the sides of a 45-45-90 triangle: B /| / | 1 / | / | sqrt(2)/2 / | / | /____________| A sqrt(2)/2 C If A is the center of the unit circle, then B is on the unit circle at an angle A = 45 degrees (pi/4). The coordinates of point B are (sqrt(2)/2, sqrt(2)/2) and the trig functions are sin(45) = BC/AB = sqrt(2)/2 cos(45) = AC/AB = sqrt(2)/2 tan(45) = BC/AC = 1 Now for angles of pi/6 and pi/3, we use the 30-60-90 triangle. It isn't isosceles like the 45-45-90 triangle, but it is half of an equilateral triangle: /|\ / | \ 1 / | \ 1 / | \ / a| \ / | \ /______|______\ 1/2 1/2 The angles of the equilateral triangle are 60 degrees. The top angle is bisected, giving the 30 degree angle. And the base of the equilateral triangle is bisected, giving segments of length 1/2. That's the key. The side opposite the 30 degree angle of the 30-60-90 triangle is half the hypotenuse. Now we can use the Pythagorean Theorem to find our one unknown, as in the 45-45-90 triangle: 1^2 = a^2 + (1/2)^2 1 = a^2 + 1/4 3/4 = a^2 sqrt(3)/2 = a That gives us all the sides of a 30-60-90 triangle: B /| / | / | 1 / | / | sqrt(3)/2 / | /______| A 1/2 C If A is the center of the unit circle, then B is on the unit circle at an angle A = 60 degrees (pi/3). The coordinates of point B are (1/2, sqrt(3)/2) and the trig functions are sin(60) = cos(30) = BC/AB = sqrt(3)/2 cos(60) = sin(30) = AC/AB = 1/2 tan(60) = BC/AC = sqrt(3) tan(30) = AC/BC = sqrt(3)/3 I don't memorize numbers very well by themselves; I find it a lot easier to remember the figures I've shown and how to use Pythagoras. I couldn't tell you how many times I have re-derived these ratios. I hope this helps you. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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