91Ó°ÊÓ

For the sine and cosine functions, the domain is _________ and the range is_________.

Use the information given to write a sinusoidal equation and sketch its graph. Recall \(B=\frac{2 \pi}{P}\). $$ \text { Max: } 100, \min : 20, P=30 $$

Sketch the following functions over the indicated interval. $$ y=3 \sin (4 t-\pi) ;[0, \pi] $$

The relationship between the coefficient \(B\), the frequency \(f\), and the period \(P\) In many applications of trigonometric functions, the equation \(y=A \sin (B t)\) is written as \(y=A \sin [(2 \pi f) t]\), where \(B=2 \pi f\). Justify the new equation using \(f=\frac{1}{P}\) and \(P=\frac{2 \pi}{B}\). In other words, explain how \(A \sin (B t)\) becomes \(A \sin [(2 \pi f) t]\), as though you were trying to help another student with the ideas involved.

In Oslo, Norway, the number of hours of daylight reaches a low of \(6 \mathrm{hr}\) in January, and a high of nearly \(18.8 \mathrm{hr}\) in July. (a) Find a sinusoidal equation model for the number of daylight hours each month; (b) sketch the graph; and (c) approximate the number of days each year there are more than \(15 \mathrm{hr}\) of daylight. Use 1 month \(\approx 30.5\) days. Assume \(t=0\) corresponds to January 1 .

Clearly state the amplitude and period of each function, then match it with the corresponding graph. $$ y=2 \sin (4 t) $$

Clearly state the amplitude and period of each function, then match it with the corresponding graph. $$ f(t)=\frac{3}{4} \cos (0.4 t) $$

Graph each function over the interval indicated, noting the period, asymptotes, zeroes, and value of \(A\) and \(B\). $$ f(t)=2 \cot (\pi t) ;[-1,1] $$

Clearly state the period of each function, then match it with the corresponding graph. $$ y=2 \csc \left(\frac{1}{2} t\right) $$

The graph of \(y=\sec x\) is shifted to the right \(\frac{\pi}{3}\) units. What is the equation of the shifted graph?