circular function

• A periodic function is a function f such that 𝑓(𝑥) = 𝑓(𝑥 + 𝑛𝑝), for every real number x in the domain of f, every integer n, and some positive real number p.
• The smallest possible positive value of p for a periodic function is the period of the function.
• The circumference of the unit circle is 2 𝜋, so the smallest value of p for which the sine and cosine functions repeat is 2 𝜋.
• The sine and cosine functions are periodic with period 2 𝜋.
• Graphing sine and cosine functions involves selecting key values of x and finding the corresponding values of sin x or cos x.
• The graph of sine function f(x) = sin x has a domain of (− ∞, ∞) and a range of [ - 1, 1], is continuous over its entire domain, and its x-intercepts are of the form (2 �� + 1) ��2, where n is an integer.
• The key points can be joined with a smooth curve, approaching the vertical asymptotes, to obtain the graph of the sine function.
• The function can be evaluated at these x-values to obtain the key points (−𝜋 2 , 3), (0,0), (�� 2 , −3).
• The maximum average monthly temperature in New Orleans is 82°F and the minimum is 54°F.
• The average monthly temperatures in New Orleans can be modeled with a sine curve.
• The vertical asymptotes of the sine function can be sketched as 𝑥 = ±�� 4.
• The two (2) asymptotes of the sine function have equations 𝑥 = −�� 4 and 𝑥 = �� 4.
• The coefficient -3 is negative, so the graph of the sine function is reflected across the x-axis compared to the graph of 𝑦 = tan 𝑥.
• The interval (−𝜋 4 , 𝜋 4 ) can be divided into four (4) equal parts, yielding key x-values of −�� 8 , 0, and �� 8.
• The graph of cosine function f(x) = cos x has a domain of (− ∞, ∞) and a range of [-1, 1], is continuous over its entire domain, and its x-intercepts are of the form (2 �� + 1) ��2, where n is an integer.
• The graph of y = 2 sin x, and compare to the graph of y = sin x.
• Sketch the graph of y = 3 2 csc (x − π 2) by drawing the typical U-shaped branches between adjacent asymptotes.
• Sketch the vertical asymptotes, which will have equations of the form x = k where k is an x-intercept of the graph of the guide function.
• Graph the corresponding reciprocal function as a guide, using a dashed curve.
• An additional period is graphed as seen in the figure on the next page.
• The branches will be above the graph of the guide function when the guide function values are positive and below the graph of the guide function when the guide function values are negative.
• The guide function to graph is y = 2𝑠𝑒𝑐 1 2 𝑥.
• Dividing this interval into four (4) equal parts gives the key points (0, 2), (π, 0), (3 π, 0), (4 π, 2) which are joined with a smooth dashed curve to indicate that this graph is only a guide.
• The guide function to graph is y = 3 2 sin (�� − 𝜋 2 ), shown as red dashed curve in the figure below.
• This guide function has amplitude 2 and one (1) period of the graph lies along the interval that satisfies the inequality 0 ≤ 1 2 𝑥 ≤ 2𝜋, or [0, 4 π ].
• To graph y = α csc bx or y = α sec bx, with b > 0, follow these steps:
• Sketch the vertical asymptotes through the x-intercepts of the graph of y = 3 2 sin (𝑥 − �� 2 ), which have the form 𝑥 = (2𝑛 + 1) 𝜋 2, where n is an integer.
• Sketch the graph of the desired function by drawing the typical U-shaped branches between the adjacent asymptotes.
• The graph of y = α sin x or y = α cos x, with α ≠ 0, will have the same shape as the graph of y = sin x or y = cos x, respectively, except with the range [- |α|, |α|].
• The amplitude of a graph of y = α sin x or y = α cos x is |α|.
• When t=0, then the beacon is aimed at point R.
• The distance d is given by 𝑑 = 4 tan 2��𝑡 where t is the time measured in seconds since the beacon started rotating.
• When the beacon is aimed to the right of R, the value of d is positive; d is negative when the beacon is aimed to the left of R.
• Let x represent the month, with January corresponding to x = 1.
• The average of the maximum and minimum temperatures is a good choice for c.
• Since the coldest month is January, when x = 1, and the hottest month is July, when x= 7, we should choose d to be about 4.
• 0.25 is a meaningless value for t because it leads to tan 𝜋 2, which is undefined.
• Using only the maximum and minimum temperatures, determine a function of the form f(x) = a sin [b(x - d)] + c, where a, b, c, and d are constants, that models the average monthly temperature in New Orleans.
• For b > 0, the graph of y = sin bx resembles that of y = sin x, but with period 2�� 𝑏.
• To graph y = α sin bx or y = α cos bx, follow these steps: find the period, 2𝜋 𝑏; start at 0 on the x-axis, and lay off a distance of 2�� 𝑏; divide the interval into four (4) equal parts; evaluate the function for each of the five (5) x-values resulting from the division; plot the points and join them with a sinusoidal curve having amplitude | α|.