414 CHAPTER 6 Trigonometric Functions NOTE If <y 0 , then =− y y , so ≥ y 1 1 means − ≥ y 1 1 or ≤− y 1 1 . If >y 0 , then = y y , so ≥ y 1 1 means ≥ y 1 1 . j Next we determine the range of each of the six trigonometric functions. Refer again to Figure 38 on the previous page. Let ( ) = P x y , be the point on the unit circle that corresponds to the angle θ. It follows that − ≤ ≤ x 1 1 and − ≤ ≤ y 1 1. Since θ = y sin and θ = x cos , we have θ θ − ≤ ≤ − ≤ ≤ 1 sin 1 1 cos 1 θ θ ≤− ≥ csc 1 or csc 1 θ θ ≤− ≥ sec 1 or sec 1 tan cot θ θ −∞< <∞ −∞< <∞ The range of both the sine function and the cosine function consists of all real numbers between −1 and 1, inclusive. Using absolute value notation, we have θ ≤ sin 1 and θ ≤ cos 1. If θ is not an integer multiple of π( )° 180 , then θ = y csc 1 . Since θ = y sin and θ = ≤ y sin 1, it follows that y y y csc 1 sin 1 1 1 1 or 1 1 . θ θ = = ≥ ≤− ≥ ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ Since θ = y csc 1 , the range of the cosecant function consists of all real numbers less than or equal to −1 or greater than or equal to 1. That is, If θ is not an odd integer multiple of π ( )° 2 90 , then θ = x sec 1 . Since θ = x cos and θ = ≤ x cos 1, it follows that θ θ = = ≥ x sec 1 cos 1 1 x x 1 1 or 1 1 ( ) ≤− ≥ . Since θ = x sec 1 , the range of the secant function consists of all real numbers less than or equal to −1 or greater than or equal to 1. That is, The range of both the tangent function and the cotangent function is the set of all real numbers. You are asked to prove this in Problems 123 and 124. Table 4 summarizes these results. Function Symbol Domain Range sine θ θ ( ) = f sin All real numbers All real numbers from −1 to 1, inclusive cosine θ θ ( ) = f cos All real numbers All real numbers from −1 to 1, inclusive tangent θ θ ( ) = f tan All real numbers, except odd integer multiples of π ( )° 2 90 All real numbers cosecant θ θ ( ) = f csc All real numbers, except integer multiples of π( )° 180 All real numbers less than or equal to −1 or greater than or equal to 1 secant θ θ ( ) = f sec All real numbers, except odd integer multiples of π ( )° 2 90 All real numbers less than or equal to −1 or greater than or equal to 1 cotangent f cot θ θ ( ) = All real numbers, except integer multiples of 180 π( )° All real numbers Table 4 Now Work PROBLEM 9 7
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