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Chapter 8: Problem 41
Use the addition formulas for tangent to simplify each expression. $$\frac{\tan t+\tan 2 t}{1-\tan t \tan 2 t}$$
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Determine all of the solutions in the interval \(0^{\circ} \leq \theta<360^{\circ}\). $$\sin \theta=\cos (\theta / 2) . \text { Hint: } \sin \theta=2 \sin (\theta / 2) \cos (\theta / 2)$$
Use graphs to determine whether there are solutions for each equation in the interval \([0,1] .\) If there are solutions, use the graphing utility to find them accurately to two decimal places. $$\cos ^{-1} x=x^{2}$$
Solve the equations on the interval \([0,2 \pi]\) as follows. Graph the expression on each side of the equation and then zoom in on the intersection points until you are certain of the first three decimal places in each answer. For instance, for Exercise \(53,\) when you graph the two equations \(y=\cos x\)and \(y=0.623\) on the interval \([0,2 \pi],\) you 'll see that there are two intersection points. The \(x\) -coordinates of these points are roots of the equation \(\cos x=0.623\). $$\cos (\sin x)=\sin x$$
(a) Use your calculator to evaluate the expression \(\cos 72^{\circ} \cos 144^{\circ} .\) Then follow steps (b) through (d) to prove that \(\cos 72^{\circ} \cos 144^{\circ}=-1 / 4\) (b) Multiply the expression \(\cos 72^{\circ} \cos 144^{\circ}\) by the quantity \(\left(\sin 72^{\circ}\right) /\left(\sin 72^{\circ}\right),\) which equals \(1 .\) Show that the result can be written $$\frac{\frac{1}{2} \sin 144_{i} \cos 144_{i}}{\sin 72_{i}}$$ (c) Explain why the expression obtained in part (b) is equal to $$\frac{\frac{1}{4} \sin 288_{i}}{\sin 72_{i}}$$ (d) Use the reference-angle concept to explain why the expression in part (c) is equal to \(-1 / 4,\) as required.
Solve the equations on the interval \([0,2 \pi]\) as follows. Graph the expression on each side of the equation and then zoom in on the intersection points until you are certain of the first three decimal places in each answer. For instance, for Exercise \(53,\) when you graph the two equations \(y=\cos x\)and \(y=0.623\) on the interval \([0,2 \pi],\) you 'll see that there are two intersection points. The \(x\) -coordinates of these points are roots of the equation \(\cos x=0.623\). $$\cos (x / 2)=\cos \left(x^{2} / 12\right)$$
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