SECTION 6.3 Properties of the Trigonometric Functions 415 2 Determine the Period of the Trigonometric Functions Look at Figure 39.This figure shows that for an angle of π 3 radians the corresponding point P on the unit circle is ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 1 2 , 3 2 . Notice that, for an angle of 3 2 radians, π π + the corresponding point P on the unit circle is also ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 1 2 , 3 2 . Then sin 3 3 2 and sin 3 2 3 2 cos 3 1 2 and cos 3 2 1 2 π π π π π π ( ) ( ) = + = = + = This example illustrates a more general situation. For a given angle θ, measured in radians, suppose that we know the corresponding point ( ) = P x y , on the unit circle. Now add π2 to θ. The point on the unit circle corresponding to θ π +2 is identical to the point P on the unit circle corresponding to θ. See Figure 40. The values of the trigonometric functions of θ π +2 are equal to the values of the corresponding trigonometric functions of θ. If we add (or subtract) integer multiples of π2 to θ, the values of the sine and cosine function remain unchanged. That is, for all θ Figure 39 π π π π π π ( ) ( ) + = + = sin 3 2 sin 3 ; cos 3 2 cos 3 y x P 5 , 1 – 2 3 –– 2 21 1 21 1 p– 3 1 2p p– 3 Figure 40 θ π θ θ π θ ( ) ( ) + = + = k k sin 2 sin ; cos 2 cos y x 21 1 21 1 P 5 (x, y) u 1 2p u In Words Tangent and cotangent have period π ; the others have period π2 . θ π θ θ π θ ( ) ( ) + = + = k k sin 2 sin cos 2 cos (1) where k is any integer. Functions that exhibit this kind of behavior are called periodic functions . Based on equation (1), the sine and cosine functions are periodic. In fact, the sine and cosine functions have period π2 .You are asked to prove this fact in Problems 125 and 126. The secant and cosecant functions are also periodic with period π2 , and the tangent and cotangent functions are periodic with period π. You are asked to prove these statements in Problems 127 through 130. These facts are summarized as follows: Because the sine, cosine, secant, and cosecant functions have period π2 , once we know their values over an interval of length π2 , we know all their values; similarly, since the tangent and cotangent functions have period π, once we know their values over an interval of length π, we know all their values. DEFINITION Periodic Function and Fundamental Period A function f is called periodic if there is a positive number p with the property that whenever θ is in the domain of f, so is θ + p, and θ θ ( ) ( ) + = f p f If there is a smallest such number p , this smallest value is called the (fundamental) period of f. Periodic Properties θ π θ θ π θ θ π θ θ π θ θ π θ θ π θ ( ) ( ) ( ) ( ) ( ) ( ) + = + = + = + = + = + = sin 2 sin cos 2 cos tan tan csc 2 csc sec 2 sec cot cot
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