By Kirkwood J.R.
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The correct answer is any of them. You can choose any full period to determine your constants. Note that the constants will depend upon which period you choose but they will all correspond to the same curve. Example 3 Given the following function ﬁnd the amplitude, vertical shift, period and horizontal shift. Then use these values to graph one period of the function. y = 2 sin πx + π −3 3 Answer: From the equation we can read oﬀ the amplitude, which is a = 2 and the vertical shift which is d = −3.
We know that if two angles are co-terminal they will have the same values for the trigonometric functions, for example sin(x + 2π) = sin(x). In particular the trigonometric functions are repeating. To make a graph of the trigonometric function we only need to determine what it looks like on an interval that contains a complete revolution. Once we have that we just copy it over and over to get the complete graph for the function. Functions that have this property are called periodic and the minimum amount of time it takes to repeat is the period.
3 θ 4 Solution First, we can use the Pythagorean theorem to ﬁnd the length of the hypotenuse. Since we have that the adjacent side has √ length 4 the opposite side has length 3 then the hypotenuse has length 32 + 42 = 5. Using the deﬁning ratios we get, 3 , 5 5 csc(θ) = , 3 sin(θ) = 4 cos(θ) = , 5 5 sec(θ) = , 4 3 tan(θ) = , 4 4 cot(θ) = . 3 The trigonometric functions take an angle and return a value. But there is more then one way to measure an angle, and 1◦ is not the same angle as 1 rad. So when you are working a problem using a calculator make sure that your calculator is in the correct angle mode.
An introduction to analysis by Kirkwood J.R.