91Ó°ÊÓ

In Exercises 83-86, determine whether each statement is true or false. $$ A \sin \left(\omega t+\frac{\omega \pi}{2}\right)=A \cos (\omega t) $$

In Exercises 83-86, assume \(A\) and \(B\) are positive real numbers. $$ \text { Find the } x \text {-intercepts of the function } y=A \sin (B x) \text {. } $$

In Exercises 83-86, assume \(A\) and \(B\) are positive real numbers. $$ \text { Find the } x \text {-intercepts of the function } y=A \cos (B x) \text {. } $$

In Exercises 83-86, determine whether each statement is true or false. $$ A \sin \left(\omega t+\frac{\omega \pi}{2}\right)=A \cos (\omega t) $$

Sketch the graph of each of the following on the same set of axes over the interval \(0 \leq x \leq 2 \pi: y=2 \sin x\) and \(y=\cos (2 x)\). Then sketch the graph of the equation \(y=2 \sin x+\cos (2 x)\) by combining the \(y\)-coordinates of the two original graphs.

Sketch the graph of each of the following on the same set of axes over the interval \(0 \leq x \leq 2 \pi\) : \(y=2 \cos x\) and \(y=2 \sin \left(\frac{1}{2} x\right)\). Then sketch the graph of the equation \(y=2 \cos x+2 \sin \left(\frac{1}{2} x\right)\) by combining the \(y\)-coordinates of the two original graphs.

Sketch the graph of each of the following on the same set of axes over the interval \(0 \leq x \leq 2 \pi: y=x\) and \(y=2 \sin x\). Then sketch the graph of the equation \(y=x+2 \sin x\) by combining the \(y\)-coordinates of the two original graphs.

If asked to graph \(y=k+A \cos (\omega t-\phi)\) over one period, state the interval for \(x\) over which the graph could be drawn [phase shift, phase shift \(+\) period].

If asked to graph \(y=k+A \cos (\omega t+\phi)\) over one period, state the interval for \(x\) over which the graph could be drawn [phase shift, phase shift \(+\) period].

Sketch the graph of each of the following on the same set of axes over the interval \(0 \leq x \leq 2 \pi: y=\frac{1}{2} x\) and \(y=-\cos x\). Then sketch the graph of the equation \(y=\frac{1}{2} x-\cos x\) by combining the \(y\)-coordinates of the two original graphs.