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Trigonometric Functions of Real Numbers; Periodic Functions • • • • Use a unit circle to define trigonometric functions of real numbers. Recognize the domain and range of sine and cosine functions Use even and odd trigonometric functions Use periodic functions Dr .Hayk Melikyan / Departmen of Mathematics and CS [email protected] H.Melikian/1200 1 The Unit Circle A unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this circle is x 2 y 2 r 2 . H.Melikian/1200 2 The Unit Circle In a unit circle, the radian measure of the central angle is equal to the length of the intercepted arc. Both are given by the same real number t. H.Melikian/1200 3 Definition: The Value of a trigonometric function at the real number t is its value at an angle of t radians If t is a real number and P = (x y) is a point on the unit circle that corresponds to t, then Unit circle y Q = (0, 1) y sin t y cos t x tan t x 1 1 x csc t sec t cot t y x y /2 x (1, 0) P(x, y) x 2 + y2 = 1 H.Melikian/1200 4 Example: Finding Values of the Trigonometric Functions Use the figure to find the values of the trigonometric functions at t. 1 sin t y 2 3 cost x 2 1 1 1 y 2 tan t x 3 3 3 2 H.Melikian/1200 3 3 3 3 5 Example: Finding Values of the Trigonometric Functions Use the figure to find the values of the trigonometric functions at t. 1 1 csct 2 y 1 2 1 1 2 2 sect x 3 3 3 2 3 2 3 3 3 3 x cot t 2 3 1 y 2 H.Melikian/1200 6 The Domain and Range of the Sine and Cosine Functions H.Melikian/1200 7 Text Example Use the figure at the right to find the values of the trigonometric functions at t = /2. y Solution The point P on the unit circle that corresponds to t = /2 has coordinates (0, 1). We use x = 0 and y = 1 to find the values of the trigonometric functions. P = (0, 1) /2 (1, 0) sin csc 2 2 y 1, cos 2 x 0 x x 2 + y2 = 1 1 1 x 0 1, cot 0 y 1 2 y 1 By definition, tan t = y/x and sec t = 1/x. Because x = 0, tan /2 and sec /2 are undefined. H.Melikian/1200 8 The Domain and Range of the Sine and Cosine Functions The domain of the sine function and the cosine function is the set of all real numbers. The range of these functions is the set of all real numbers from –1 to 1, inclusive. H.Melikian/1200 9 Even and Odd Trigonometric Functions The cosine and secant functions are even. cos (-t) = cos t sec(-t) = sec t The sine, cosecant, tangent, and cotangent functions are odd. sin (-t) = – sin t csc (-t) = – csc t tan (-t) = – tan t cot (-t) = – cot t H.Melikian/1200 10 Example Find the exact value of sin (-45º) 2 sin( 45 ) sin( 45 ) 2 o H.Melikian/1200 o 11 Definition of a Periodic Function • A function f is periodic if there exists a positive number p such that • f (t + p) = f (t) • For all t in the domain of f. The smallest number p for which f is periodic is called the period of f. H.Melikian/1200 12 Periodic Properties of the Sine and Cosine Functions The sine and cosine functions are periodic functions and have period 2. • sin (t + 2) = sin t • cos (t + 2) = cos t Example Find the exact value of cos(5) cos(5 ) cos( 4 ) cos 1 H.Melikian/1200 13 Periodic Properties of the Tangent and Cotangent Functions • tan (t + ) = tan t and cot (t + ) = cot t • The tangent and cotangent functions are periodic functions and have period . H.Melikian/1200 14 Repetitive Behavior of the Sine, Cosine, and Tangent Functions For any integer n and real number t, is sin (t + 2 n) = sin t, cos (t + 2 n) = cos t, and tan (t + n) = tan t. H.Melikian/1200 15 Home Work 1. H.Melikian/1200 16