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Chapter 5: Problem 9
Verify each identity. $$\cos ^{2} x-\sin ^{2} x=1-2 \sin ^{2} x$$
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Use this information to solve. Our cycle of normal breathing takes place every 5 seconds. Velocity of air flow, \(y,\) measured in liters per second, after \(x\) seconds is modeled by $$y=0.6 \sin \frac{2 \pi}{5} x$$ Velocity of air flow is positive when we inhale and negative when we exhale. Within each breathing cycle, when are we exhaling at a rate of 0.3 liter per second? Round to the nearest tenth of a second.
Suppose you are solving equations in the interval \([0,2 \pi)\) Without actually solving equations, what is the difference between the number of solutions of \(\sin x=\frac{1}{2}\) and \(\sin 2 x=\frac{1}{2} ?\) How do you account for this difference?
Use a sketch to find the exact value of \(\sec \left(\sin ^{-1} \frac{1}{2}\right)\).
Use a graphing utility to approximate the solutions of each equation in the interval \([0,2 \pi) .\) Round to the nearest hundredth of a radian. $$\sin x+\sin 2 x+\sin 3 x=0$$
Use this information to solve. When throwing an object, the distance achieved depends on its initial velocity, \(v_{0}\) and the angle above the horizontal at which the object is thrown, \(\theta\) The distance, \(d\), in feet, that describes the range covered is given by $$d=\frac{v_{0}^{2}}{16} \sin \theta \cos \theta$$ where \(v_{0}\) is measured in feet per second. You and your friend are throwing a baseball back and forth. If you throw the ball with an initial velocity of \(v_{0}=90\) feet per second, at what angle of elevation, \(\theta,\) to the nearest degree, should you direct your throw so that it can be easily caught by your friend located 170 feet away?
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