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the unit circle the 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 since x2 x2 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 know more about right triangle trigonometry tutorcircle.com page no 1/4

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p. 2

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 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 px1,y1 on the unit circle such that an angle t with 0 t /2 is formed with the positive arm of the x-axis now consider a point qx1,0 and line segments pq oq the result is a right triangle opq with qop t because pq has length y1 oq length x1 and oa length 1 sint y1 and cost 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 os the result is a right triangle ors with sor t it can hence be seen that because roq -t r is at cos t sin t in the same way that p is at cost sint the conclusion is that since x1,y1 is the same as cos t sin t and x1,y1 is the same as cost sint it is true that sint sin t and -cost cos t it may be inferred in a similar manner that tan t -tant since tant y1/x1 and tan t y1 x1 a simple demonstration of the above can be seen in the equality sin 4 sin3/4 1/sqrt2 when working with right triangles sine cosine and other trigonometric functions only make sense for angle measures more than zero and less than /2 read more about circle chord calculator tutorcircle.com page no 2/4

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however when defined with the unit circle these functions produce meaningful values for any real-valued angle measure ­ even those greater than 2 in fact all six standard trigonometric functions ­ sine cosine tangent cotangent secant and cosecant as well as archaic functions like versine and exsecant ­ 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 tutorcircle.com page no 3/4 page no 2/3

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p. 4

thank you for watching presentation

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