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deriving the unit circle

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From Wikipedia

Unit circle

In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S1; the generalization to higher dimensions is the unit sphere.

If (x, y) is a point on the unit circle in the first quadrant, then x and y are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation

x^2 + y^2 = 1.

Since x2 = (−x)2 for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not just those in the first quadrant.

One may also use other notions of "distance" to define other "unit circles", such as the Riemannian circle; see the article on mathematical norms for additional examples.

Forms of unit circle points

  • exponential :
z = \,\mathrm{e}^{i t}\,
  • trigonometric :
z = \cos(t) + i \sin(t) \,

Trigonometric functions on the unit circle

The trigonometric functions cosine and sine may be defined on the unit circle as follows. If (x, y) is a point of the unit circle, and if the ray from the origin (0, 0) to (x, y) makes an anglet from the positive x-axis, (where counterclockwise turning is positive), then

\cos(t) = x \,\!
\sin(t) = y. \,\!

The equation x2 + y2 = 1 gives the relation

\cos^2(t) + \sin^2(t) = 1. \,\!

The unit circle also demonstrates that sine and cosine are periodic functions, with the identities

\cos t = \cos(2\pi k+t) \,\!
\sin t = \sin(2\pi k+t) \,\!

for any integerk.

Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OA from the origin to a point P(x1,y1) on the unit circle such that an angle t with 0 < t< π/2 is formed with the positive arm of the x-axis. Now consider a point Q(x1,0) and line segments PQ \perp OQ. The result is a right triangle ΔOPQ with ∠QOP = t. Because PQ has length y1, OQ length x1, and OA length 1, sin(t) = y1 and cos(t) = x1. Having established these equivalences, take another radius OR from the origin to a point R(−x1,y1) on the circle such that the same angle t is formed with the negative arm of the x-axis. Now consider a point S(−x1,0) and line segments RS \perp OS. The result is a right triangle ΔORS with ∠SOR = t. It can hence be seen that, because ∠ROQ = π−t, R is at (cos(π−t),sin(π−t)) in the same way that P is at (cos(t),sin(t)). The conclusion is that, since (−x1,y1) is the same as (cos(π−t),sin(π−t)) and (x1,y1) is the same as (cos(t),sin(t)), it is true that sin(t) = sin(π−t) and −cos(t) = cos(π−t). It may be inferred in a similar manner that tan(π−t) = −tan(t), since tan(t) = y1/x1 and tan(π−t) = y1/(−x1). A simple demonstration of the above can be seen in the equality sin(π/4) = sin(3π/4) = 1/sqrt(2).

When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the unit circle, these functions produce meaningful values for any real-valued angle measure&nbsp;– even those greater than 2π. In fact, all six standard trigonometric functions&nbsp;– sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like versine and exsecant&nbsp;– can be defined geometrically in terms of a unit circle, as shown at right.

Using the unit circle, the values of any trigonometric function for many angles other than those labeled can be calculated without the use of a calculator by using the Sum and Difference Formulas.

Circle group

Complex numbers can be identified with points in the Euclidean plane, namely the number a + bi is identified with the point (a, b). Under this identification, the unit circle is a group under multiplication, called the circle group. This group has important applications in mathematics and science.

Complex dynamics

Julia set of discrete nonlinear dynamical system with evolution function:

f_0(x) = x^2 \,

is a unit circle. It is a simplest case so it is widely used in study of dynamical systems.

From Yahoo Answers

Question:In my high school trig. class we were given a unit circle to memorize for a test (it has 16 points, with each point's angle measure in degrees/radians, and coordinates). I can get by on this test by just figuring out those characteristics in different ways when i see the problems (without memorizing it before hand) I was wondering though, if i will need to know that stuff for later math classes. In other words, is it worth knowing?

Answers:what math teacher today do not do when they teach is tell you what is it good for in real life..... rarely do teacher ever do that and sometimes we, student lose the our interest or goal as in what reward will this bring but all the math class as in , algebra, geometry, trig, all are foundation of calculus.... which is thinking math, math that requires thoughts instead of just repetative work...

Question:I need to find a way to easily remember how to plot the radians on the unit circle..i would convert them , but that would take too long and my teacher doesnt want us to do that.

Answers:You just have to remember one fact: Going all the way around a circle is 2pi radians. You can derive everything from that quickly. 0 = 0 90 = pi/2 180 = pi 270 = 3pi/2 360 = 2pi

Question:? And what does a liter measure? Volume right?

Answers:A derived unit is obtained by combining base units by multiplication, division or both of these operations. It's units is derived from a similar combination of base units. Hence, volume = length x length x length = metre x metre x metre = m^3 See, multiplication of the same base units, metre gives you volume.

Question:I read the whole chap. On it and still can't figure it out. Its chemistry

Answers:It's not really chemistry. It's just a way of measurement. Chemistry has some, physics has others, other sciences have still more. A standard unit is one that is one of the SI units, such as time(seconds), distance(meters), mass(kilograms)... Derived units use a combination, such as force((kilogram*meters)/(second*second) -> Newton).

From Youtube

Unit Circle Pt6: Deriving a Trig Identity sin^2+cos^2=1: Trigonometry :123MrBee's Trigonometry Playlist: www.youtube.com Please comment, rate, and subscribe!- Mr B

circle derivation 3 :to derive the equation of a tangent of a circle