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deriving the unit circle
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From Wikipedia
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 S^{1}; 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 x^{2} = (−x)^{2} for all x, and since the reflection of any point on the unit circle about the x or yaxis 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 xaxis, (where counterclockwise turning is positive), then
 \cos(t) = x \,\!
 \sin(t) = y. \,\!
The equation x^{2} + y^{2} = 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(x_{1},y_{1}) on the unit circle such that an angle t with 0 < t< Ï€/2 is formed with the positive arm of the xaxis. Now consider a point Q(x_{1},0) and line segments PQ \perp OQ. The result is a right triangle Î”OPQ with âˆ QOP = t. Because PQ has length y_{1}, OQ length x_{1}, and OA length 1, sin(t) = y_{1} and cos(t) = x_{1}. Having established these equivalences, take another radius OR from the origin to a point R(âˆ’x_{1},y_{1}) on the circle such that the same angle t is formed with the negative arm of the xaxis. Now consider a point S(âˆ’x_{1},0) and line segments RS \perp OS. The result is a right triangle Î”ORS with âˆ SOR = t. It can hence be seen that, because âˆ ROQ = Ï€âˆ’t, R is at (cos(Ï€âˆ’t),sin(Ï€âˆ’t)) in the same way that P is at (cos(t),sin(t)). The conclusion is that, since (âˆ’x_{1},y_{1}) is the same as (cos(Ï€âˆ’t),sin(Ï€âˆ’t)) and (x_{1},y_{1}) is the same as (cos(t),sin(t)), it is true that sin(t) = sin(Ï€âˆ’t) and âˆ’cos(t) = cos(Ï€âˆ’t). It may be inferred in a similar manner that tan(Ï€âˆ’t) = âˆ’tan(t), since tan(t) = y_{1}/x_{1} and tan(Ï€âˆ’t) = y_{1}/(âˆ’x_{1}). A simple demonstration of the above can be seen in the equality sin(Ï€/4) = sin(3Ï€/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 Ï€/2. However, when defined with the unit circle, these functions produce meaningful values for any realvalued angle measure â€“ even those greater than 2Ï€. In fact, all six standard trigonometric functions â€“ sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like versine and exsecant â€“ 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.
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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...
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
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.
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).
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