SECTION 6.3 Properties of the Trigonometric Functions 413 1 Determine the Domain and the Range of the Trigonometric Functions Let θ be an angle in standard position, and let ( ) = P x y , be the point on the unit circle that corresponds to θ. See Figure 38. Then, by the definition given earlier, Figure 38 y x (21, 0) (1, 0) u P 5 (x, y) (0, 21) (0, 1) O θ θ θ θ θ θ = = = ≠ = ≠ = ≠ = ≠ y x y x x y y x x x y y sin cos tan 0 csc 1 0 sec 1 0 cot 0 For θ sin and θ cos , there is no concern about dividing by 0, so θ can be any angle. It follows that the domain of the sine function and cosine function is the set of all real numbers. • The domain of the sine function is the set of all real numbers. • The domain of the cosine function is the set of all real numbers. For the tangent function and secant function, the x -coordinate of ( ) = P x y , cannot be 0 since this results in division by 0. See Figure 38. On the unit circle, there are two such points, ( ) 0, 1 and ( ) − 0, 1 . These two points correspond to the angles π ( )° 2 90 and π ( )° 3 2 270 or, more generally, to any angle that is an odd integer multiple of π ( )° 2 90 , such as π ( ) ± ± ° 2 90 , π π ( ) ( ) ± ± ° ± ± ° 3 2 270 , 5 2 450 , and so on. Such angles must be excluded from the domain of the tangent function and secant function. • The domain of the tangent function is the set of all real numbers, except odd integer multiples of π ( )° 2 90 . • The domain of the secant function is the set of all real numbers, except odd integer multiples of π ( )° 2 90 . For the cotangent function and cosecant function, the y -coordinate of ( ) = P x y , cannot be 0 since this results in division by 0. See Figure 38. On the unit circle, there are two such points, ( ) 1, 0 and ( ) −1, 0 . These two points correspond to the angles ( )° 00 and π( )° 180 or, more generally, to any angle that is an integer multiple of π( )° 180 , such as ( )° 00 , π π ( ) ( ) ± ± ° ± ± ° 180, 2 360, π( ) ± ± ° 3 540 , and so on. Such angles must therefore be excluded from the domain of the cotangent function and cosecant function. • The domain of the cotangent function is the set of all real numbers, except integer multiples of π( )° 180 . • The domain of the cosecant function is the set of all real numbers, except integer multiples of π( )° 180 .
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