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Chapter 7: Problem 39
Write each expression as a product of trigonometric functions. $$\cos 4 x-\cos 2 x$$
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In an electric circuit, suppose that \(V=\cos 2 \pi t\) models the electromotive force in volts at \(t\) seconds. Find the least value of \(t\) where \(0 \leq t \leq \frac{1}{2}\) for each value of \(V\) (a) \(V=0\) (b) \(V=0.5\) (c) \(V=0.25\)
Verify that each trigonometric equation is an identity. $$\frac{1-\cos x}{1+\cos x}=\csc ^{2} x-2 \csc x \cot x+\cot ^{2} x$$
(This discussion applies to functions of both angles and real numbers.) Consider the following. $$\begin{aligned}\cos (&\left.180^{\circ}-\theta\right) \\\&=\cos 180^{\circ} \cos \theta+\sin 180^{\circ} \sin \theta \\\&=(-1) \cos \theta+(0) \sin \theta \\\&=-\cos \theta\end{aligned}$$ \(\cos \left(180^{\circ}-\theta\right)=-\cos \theta\) is an example of a reduction formula, which is an identity that reduces a function of a quadrantal angle plus or minus \(\theta\) to a function of \(\theta\) alone. Another example of a reduction formula is \(\cos \left(270^{\circ}+\theta\right)=\sin \theta\) Here is an interesting method for quickly determining a reduction formula for a trigonometric function \(f\) of the form \(f(Q \pm \theta),\) where \(Q\) is a quadrantal angle. There are two cases to consider, and in each case, think of \(\boldsymbol{\theta}\) as a small positive angle in order to determine the quadrant in which \(Q \pm \theta\) will lie. Case 1 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along the \(x\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a small positive angle \(\theta .\) If the given function \(f\) is positive in that quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that quadrant, use a - sign. The reduced form will have that \(\operatorname{sign}, f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY THE GRAPH Case 2 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along the \(y\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a small positive angle \(\theta .\) If the given function \(f\) is positive in that quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that quadrant, use a - sign. The reduced form will have that sign, the cofunction of \(f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY THE GRAPH Use these ideas to write reduction formulas for each of the following. $$\sin \left(180^{\circ}+\theta\right)$$
Consider the function $$f(x)=3 x-2 \quad \text { and its inverse } \quad f^{-1}(x)=\frac{1}{3} x+\frac{2}{3}$$. Simplify \(f\left(f^{-1}(x)\right)\) and \(f^{-1}(f(x)) .\) What do you notice in each case? What would the graph look like in each case?
Use identities to write each expression as a single function of \(x\) or \(\theta\). $$\sin \left(270^{\circ}-\theta\right)$$
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