398 CHAPTER 6 Trigonometric Functions Notice in these definitions that if x 0, = that is, if the point P is on the y -axis, then the tangent function and the secant function are undefined.Also, if y 0, = that is, if the point P is on the x -axis, then the cosecant function and the cotangent function are undefined. Because we use the unit circle in these definitions of the trigonometric functions, they are sometimes referred to as circular functions . 1 Find the Exact Values of the Trigonometric Functions Using a Point on the Unit Circle DEFINITION Trigonometric Functions of a Real Number Let t be a real number and let P x y , ( ) = be the point on the unit circle that corresponds to t. The sine function associates with t the y -coordinate of P and is denoted by t y sin = The cosine function associates with t the x -coordinate of P and is denoted by t x cos = If x 0, ≠ the tangent function associates with t the ratio of the y -coordinate to the x -coordinate of P and is denoted by t y x tan = If y 0, ≠ the cosecant function is defined as t y csc 1 = If x 0, ≠ the secant function is defined as t x sec 1 = If y 0, ≠ the cotangent function is defined as t x y cot = In Words The point ( ) =P x y , on the unit circle corresponding to a real number t is given by ( ) t t cos , sin . Finding the Values of the Six Trigonometric Functions Using a Point on the Unit Circle Let t be a real number and let P 1 2 , 3 2 = − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ be the point on the unit circle that corresponds to t. Find the values of t t t t t sin , cos , tan , csc , sec , and t cot . EXAMPLE 1 Solution See Figure 21.We follow the definition of the six trigonometric functions, using P x y 1 2 , 3 2 , . ( ) = − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = Then, with x y 1 2 and 3 2 , = − = we have = = = = − = = − = − = = = = = − = − = = − = − t y t x t y x t y t x t x y sin 3 2 cos 1 2 tan 3 2 1 2 3 csc 1 1 3 2 2 3 3 sec 1 1 1 2 2 cot 1 2 3 2 3 3 Figure 21 y x t P 5 , (1, 0) 1 – 2 3 –– 2 2 CAUTION When writing the values of the trigonometric functions, do not forget the argument of the function. t sin 3 2 sin 3 2 = = correct incorrect j Now Work PROBLEM 13
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