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Chapter 5: Problem 42
In Exercises \(37-42,\) find the exact value of the expression. $$ \frac{\tan 25^{\circ}+\tan 110^{\circ}}{1-\tan 25^{\circ} \tan 110^{\circ}} $$
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In Exercises 91-98, use the sum-to-product formulas to write the sum or difference as a product. \( \sin 3 \theta + \sin \theta \)
In Exercises 37-42, find the exact values of \( \sin 2u \), \( \cos 2u \), and \( \tan 2u \) using the double-angle formulas. \( \tan u = \dfrac{3}{5}, 0 < u < \dfrac{\pi}{2} \)
In Exercises 73-76, use the half-angle formulas to simplify the expression. \( \sqrt{\dfrac{1 - \cos 6x}{2}} \)
The mach number \(M\) of an 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 (see figure). The mach number is related to the apex angle \(\theta\) of the cone by \(\sin (\theta / 2)=1 / M.\) (Figure Cant Copy) (a) Find the angle \(\theta\) that corresponds to a mach number of \(1 .\) (b) Find the angle \(\theta\) that corresponds to a mach number of \(4.5 .\) (c) The speed of sound is about 760 miles per hour. Determine the speed of an object with the mach numbers from parts (a) and (b). (d) Rewrite the equation in terms of \(\theta\)
In Exercises 129 and 130, graph the function by hand in the interval \(\left[0,2\pi\right] \) by using the power-reducing formulas. \( f(x) = \sin^2 x \)
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