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Chapter 5: Problem 85
Determine whether the statement is true or false. Justify your answer. $$\sin 60^{\circ} \csc 60^{\circ}=1$$
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(a) Use a graphing utility to complete the table. $$\begin{array}{|l|l|l|l|l|l|l|} \hline \theta & 0 & 0.3 & 0.6 & 0.9 & 1.2 & 1.5 \\ \hline \cos \left(\frac{3 \pi}{2}-\theta\right) & & & & & & \\ \hline-\sin \theta & & & & & & \\ \hline \end{array}$$ (b) Make a conjecture about the relationship between \(\cos \left(\frac{3 \pi}{2}-\theta\right)\) and \(-\sin \theta\).
Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=5 \pi / 2$$
A buoy oscillates in simple harmonic motion as waves go past. The buoy moves a total of 3.5 feet from its high point to its low point (see figure), and it returns to its high point every 10 seconds. Write an equation that describes the motion of the buoy, where the high point corresponds to the time \(t=0.\)
In calculus, it is shown that the area of the region bounded by the graphs of \(y=0, y=1 /\left(x^{2}+1\right), x=a,\) and \(x=b\) is given by Arca \(=\arctan b-\arctan a\) (see figure). Find the area for each value of \(a\) and \(b\) (a) \(a=0, b=1\) (b) \(a=-1, b=1\) (c) \(a=0, b=3\) (d) \(a=-1, b=3\)
Define the inverse cosecant function by restricting the domain of the cosecant function to the intervals \([-\pi / 2,0)\) and \((0, \pi / 2],\) and sketch the graph of the inverse function.
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