what is a way to memorize the unit circle easily...suggestions??

Question

the unit circle as in what's pi/3= (1/2, (root (3) /2)), pi/4, etc??

Answer 1

Yes. It's actually a very simple sequence. Consider the values of the sine function.

sin(0) = 0, sin(pi/6) = 1/2, sin(pi/4) = sqr(2)/2, sin(pi/3) = sqr(3)/2, sin(pi/2) = 1, sin(2pi/3) = sqr(3)/2, sin(3pi/4) = sqr(2)/2, sin(5pi/6) = 1/2. etc...

Think of it like this.

sin(0) = sqr(0)/2, sin(pi/6) = sqr(1)/2, sin(pi/4) = sqr(2)/2, sin(pi/3) = sqr(3)/2, sin(pi/2) = sqr(4)/2, sqr(2pi/3) = sqr(3)/2, sin(3pi/4) = sqr(2)/2, sin(5pi/6) = sqr(1)/2

As you can see all those expressions are equivalent to the ones above. For the traditional values of the unit circle it is always the square root of some integer 0-4 divided by 2. At pi/2 you simply reverse the trend. To know cosine is trivial if you know sine. They're the same at pi/4 which should be pretty intuitive. Then you know that they're the opposite of each other at values of pi that are divided by 3 or 6 in the sense if one is sqr(3)/2 then the other is 1/2. Also sine is 0 at 0 and pi while cosine is 0 at pi/2 and 3pi/2. If that is not easy to remember for you consider the length of the "opposite" in a "triangle" with an angle of 0 degrees or the "adjacent" in a "triangle" with two 90 degree angles.

What remains is to know the signs. Knowing that cosine is adjacent over hypotenuse and sine is opposite over hypotenuse just imagine a coordinate grid system starting at the center of the unit circle. The sign of sine depends on what "y" quardrant it is in and the sign of cosine depends on which "x" quadrant it is in as that corresponds to opposite and adjacent respectively. If that does not make sense there is a mnemonic you can memorize. All Students Take Calculus. Each letter goes into each quadrant counterclockwise starting with the top right quadrant. A stands for "All" meaning all functions are positive in the first quadrant (0 - pi/2). S stands for "sine" meaning just sine is positive in quadrant II (pi/2 - pi). T is for tangent in quadrant III (pi - 3pi/2). C is for cosine in quadrant IV (3pi/2-2pi). If a function is not positive then it is negative.

Knowing the signs and values of sine and cosine at every point is the essence of the unit circle. Everything else follows easily from these two functions. Hope that helped.

Answer 2

x^2+y^2=1.
.

Answer 3

Memorize the unit circle? It's simply a circle with its center at the orgin (point (0,0)) and a radius of one.

Why do you need to memorize that? I don't think I understand what you mean.

Do you mean trigonometric values of common angles within the unit circle? Please clarify.