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A rotating beacon is located at point \(A\) next to a long wall. The beacon is \(4 \mathrm{m}\) from the wall. The distance \(d\) is given by $$d=4 \tan 2 \pi t$$ where \(t\) is time measured in seconds since the beacon started rotating. (When \(t=0\) the beacon is aimed at point \(R\). 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 \(R .\) ) Find \(d\) for each time. (a) \(t=0\) (b) \(t=0.4\) (c) \(t=0.8\) (d) \(t=1.2\) (e) Why is 0.25 a meaningless value for \(t ?\) CAN'T COPY THE GRAPH

Graph each function over a one-period interval. $$y=4-3 \cos (x-\pi)$$

Find the value of \(s\) in the interval \(\left[0, \frac{\pi}{2}\right]\) that makes each statement true. $$\sec s=1.0806$$

Find the exact value of \(s\) in the given interval that has the given circular function value. Do not use a calculator. $$\left[\pi, \frac{3 \pi}{2}\right] ; \quad \tan s=\sqrt{3}$$

Over the interval \([0,2 \pi],\) compare the graphs of $$ y=\sin 2 x \quad \text { and } \quad y=2 \sin x $$ Can we say that, in general, \(\sin b x=b \sin x ?\) Explain.

Find the length to three significant digits of each arc intercepted by a central angle \(\theta\) in a circle of radius \(r\). $$r=0.892 \mathrm{cm}, \theta=\frac{11 \pi}{10} \mathrm{radians}$$

Suppose that point \(P\) is on a circle with radius \(r,\) and ray \(O P\) is rotating with angular speed \(\omega .\) For the given values of \(r, \omega,\) and \(t,\) find each of the following. (a) the angle generated by \(P\) in time \(t\) (b) the distance traveled by \(P\) along the circle in time \(t\) (c) the linear speed of \(P\) $$r=30 \mathrm{cm}, \omega=\frac{\pi}{10} \text { radian per } \sec , t=4 \mathrm{sec}$$

Radian measure simplifies many formulas, such as the formula for arc length, \(s=r \theta .\) Give the corresponding formula when \(\theta\) is measured in degrees instead of radians.

Find the distance in kilometers between each pair of cities, assuming they lie on the same north-south line. Use \(r=6400 \mathrm{km}\) for the radius of Earth. Panama City, Panama, \(9^{\circ} \mathrm{N},\) and Pittsburgh, Pennsylvania, \(40^{\circ} \mathrm{N}\)

Find the distance in kilometers between each pair of cities, assuming they lie on the same north-south line. Use \(r=6400 \mathrm{km}\) for the radius of Earth. New York City, New York, \(41^{\circ} \mathrm{N},\) and Lima, Peru, \(12^{\circ} \mathrm{S}\)