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Chapter 5: Problem 28
Write the expression as the sine, cosine, or tangent of an angle. $$\cos \frac{\pi}{7} \cos \frac{\pi}{5}-\sin \frac{\pi}{7} \sin \frac{\pi}{5}$$
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Use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. $$\sin ^{2} 2 x \cos ^{2} 2 x$$
The Mach number \(M\) of a supersonic airplane is the ratio of its speed to the speed of sound. When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane. The Mach number is related to the apex angle \(\theta\) of the cone by \(\sin (\theta / 2)=1 / M\) (a) Use a half-angle formula to rewrite the equation in terms of \(\cos \theta\). (b) Find the angle \(\theta\) that corresponds to a Mach number of 1. (c) Find the angle \(\theta\) that corresponds to a Mach number of 4.5. (d) The speed of sound is about 760 miles per hour. Determine the speed of an object with the Mach numbers from parts (b) and \((\mathrm{c})\).
Use the half-angle formulas to simplify the expression. $$-\sqrt{\frac{1-\cos 8 x}{1+\cos 8 x}}$$
Find the exact value of the trigonometric expression given that \(\sin u=-\frac{7}{25}\) and \(\cos v=-\frac{4}{5} .\) (Both \(u\) and \(v\) are in Quadrant III.) $$\tan (u-v)$$
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 \(h \rightarrow 0^{+}\).
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