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Chapter 5: Problem 7
Verify that the \(x\) -values are solutions of the equation. \(3 \tan ^{2} 2 x-1=0\) (a) \(x=\frac{\pi}{12}\) (b) \(x=\frac{5 \pi}{12}\)
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Use the sum-to-product formulas to rewrite the sum or difference as a product. $$\cos 6 x+\cos 2 x$$
Verify the identity. $$\cos ^{4} x-\sin ^{4} x=\cos 2 x$$
Use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. $$\sin ^{2} 2 x \cos ^{2} 2 x$$
A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by $$y=\frac{1}{3} \sin 2 t+\frac{1}{4} \cos 2 t$$.where \(y\) is the distance from equilibrium (in feet) and \(t\) is the time (in seconds). (A). Use the identity \(a \sin B \theta+b \cos B \theta=\sqrt{a^{2}+b^{2}} \sin (B \theta+C)\) where \(C=\arctan (b / a), a>0,\) to write the model in the form \(y=\sqrt{a^{2}+b^{2}} \sin (B t+C)\). (B) Find the amplitude of the oscillations of the weight. (C) Find the frequency of the oscillations of the weight.
Find all solutions of the equation in the interval \([0,2 \pi)\). $$\cos \left(x+\frac{\pi}{4}\right)-\cos \left(x-\frac{\pi}{4}\right)=1$$
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