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Chapter 5: Problem 44
Solve the multiple-angle equation. $$2 \sin \frac{x}{2}+\sqrt{3}=0$$
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Write the expression as the sine, cosine, or tangent of an angle. $$\frac{\tan 140^{\circ}-\tan 60^{\circ}}{1+\tan 140^{\circ} \tan 60^{\circ}}$$
Find all solutions of the equation in the interval \([0,2 \pi)\). $$\sin \left(x+\frac{\pi}{6}\right)-\sin \left(x-\frac{7 \pi}{6}\right)=\frac{\sqrt{3}}{2}$$
Let \(x=\pi / 3\) in the identity in Example 8 and define the functions \(f\) and \(g\) as follows. \begin{array}{l}f(h)=\frac{\sin [(\pi / 3)+h]-\sin (\pi / 3)}{h} \\\g(h)=\cos \frac{\pi}{3}\left(\frac{\sin h}{h}\right)-\sin \frac{\pi}{3}\left(\frac{1-\cos h}{h}\right)\end{array} (a) What are the domains of the functions \(f\) and \(g ?\) (b) Use a graphing utility to complete the table.$$\begin{array}{|l|l|l|l|l|l|l|}\hline h & 0.5 & 0.2 & 0.1 & 0.05 & 0.02 & 0.01 \\\\\hline f(h) & & & & & & \\\\\hline g(h) & & & & & & \\\\\hline\end{array}$$. (c) Use the graphing utility to graph the functions \(f\) and \(g\). (d) Use the table and the graphs to make a conjecture about the values of the functions \(f\) and \(g\) as \(h \rightarrow 0^{+}\).
Find all solutions of the equation in the interval \([0,2 \pi) .\) Use a graphing utility to graph the equation and verify the solutions. $$\sin \frac{x}{2}+\cos x=0$$
Use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. $$\sin ^{4} x \cos ^{2} x$$
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