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Chapter 6: Problem 46
Convert the given radian measure to degrees. $$-\pi / 60$$
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Use the graphs of the trigonometric functions to determine the number of solutions of the equation between 0 and \(2 \pi\) $$\sin t=-1 / 2$$
In Exercises \(61-64\), use graphs to determine whether the equa. tion could possibly be an identity or is definitely not an identity. $$\frac{\sec t+\csc t}{1+\tan t}=\csc t$$
The diagram shows a merry-go-round that is turning counterclockwise at a constant rate, making 2 revolutions in 1 minute. On the merry-go-round are horses \(A, B, C,\) and \(D\) at 4 meters from the center and horses \(E, F,\) and \(G\) at 8 meters from the center. There is a function \(a(t)\) that gives the distance the horse \(A\) is from the \(y\) -axis (this is the \(x\) -coordinate of the position \(A\) is in ) as a function of time \(t\) (measured in minutes). Similarly, \(b(t)\) gives the \(x\) -coordinate for \(B\) as a function of time, and so on. Assume that the diagram shows the situation at time \(t=0\). (Check your book to see figure) (a) Which of the following functions does \(a(t)\) equal? $$\begin{array}{ll}4 \cos t, & 4 \cos \pi t, \quad 4 \cos 2 t, \quad 4 \cos 2 \pi t \\\4 \cos \left(\frac{1}{2} t\right), & 4 \cos ((\pi / 2) t), \quad 4 \cos 4 \pi t\end{array}$$ Explain. (b) Describe the functions \(b(t), c(t), d(t),\) and so on using the cosine function: $$\begin{array}{l}b(t)=\longrightarrow(t)=\longrightarrow d(t)= \\\e(t)=\longrightarrow f(t)=\longrightarrow g(t)=\end{array}$$ (c) Suppose the \(x\) -coordinate of a horse \(S\) is given by the function \(4 \cos (4 \pi t-(5 \pi / 6))\) and the \(x\) -coordinate of another horse \(T\) is given by \(8 \cos (4 \pi t-(\pi / 3))\) Where are these horses located in relation to the rest of the horses? Mark the positions of \(T\) and \(S\) at \(t=0\) into the figure.
Graph \(f(t)\) in a viewing window with \(-2 \pi \leq t \leq 2 \pi .\) Use a maximum finder and a root finder to determine constants \(A, b, c\) such that the graph of \(f(t)\) appears to coincide with the graph of \(g(t)=A \sin (b t+c).\) $$f(t)=3 \sin t+2 \cos t$$
In Exercises \(55-60\), find the values of all six trigonometric functions at \(t\) if the given conditions are true. $$\sin t=-2 / 3 \quad \text { and } \quad \sec t>0$$
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