Section 5.3 - Section 5.3 Properties of the Trigonometric...

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Unformatted text preview: Section 5.3 Properties of the Trigonometric Functions Domain and range To see the domains and ranges it is best to consider the unit circle. Table pg.357 Domain The domain of the sin, y/r, and cos , x/r, functions are all real's. The domain tan , y/x, and sec , 1/x, functions are all real's except odd multiples of /2, i.e. 90 The domain of the csc , 1/y, and cot , x/y, functions are integral multiples of , i.e. 180 Range 1sin 1 and 1 cos 1. sec, csc 1 and sec, csc 1 tan, cot Period of the trigonometric functions Look at our picture again. From this we can see how making one full sweep of the circle gives us the same values for the trigonometric functions. Period Def: A function f is called periodic if there is a positive number p such that, whenever is in the domain of f, so is +p, and f(+p)=f(p) If there is a smallest such number p, the smallest is called the fundamental period of f. Generally the fundamental period is refered to as just the period. Periodic properties These can be seen on pg. 358 If you look at the unit circle (yet again), you can see where these come from. Signs of the trigonometric functions These come from the x and y values in the quadrants. Pg. 360 Identities sin2()+cos2()=1 Divide this by cos2() to get tan2()+1=sec2() Divide by sin2() to get cot2()+1=csc2() Finding functions when one is known Simply use the definitions to obtain the other values. EvenOdd Properties These properties, on pg 366. come directly from the unit circle. ...
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