91Ó°ÊÓ

Is the graph of \(y=\sin x\) symmetric with respect to a reflection in the origin? Justify your answer.

Is the graph of \(y=\sin 2\left(x+\frac{\pi}{2}\right)\) the same as the graph of \(y=\sin 2\left(x-\frac{\pi}{2}\right) ?\) Justify your answer.

If tan \(x\) increases for all values of \(x\) for which it is defined, explain why cot \(x\) decreases for all values of \(x\) for which it is defined.

Calvin said that the graph of \(y=\tan \left(x-\frac{\pi}{4}\right)\) has asymptotes at \(x=\frac{3 \pi}{4}+n \pi\) for all integral values of \(n .\) Do you agree with Calvin? Explain why or why not.

Show that if arcsin \(x=-\frac{1}{2},\) then the measure of the reference angle for \(x\) is \(30^{\circ} .\)

Is the graph of \(y=\cos x\) its own image under a reflection in the \(y\) -axis? Justify your answer.

List at least three ways in which the graph of the tangent function differs from the graph of the sine function and the cosine function.

Tyler said that one cycle of a cosine curve has a maximum value at \(\left(\frac{\pi}{4}, 5\right)\) and a minimum value at \(\left(\frac{5 \pi}{4},-5\right) .\) The equation of the curve is \(y=5 \cos \left(2 x-\frac{\pi}{2}\right) .\) Do you agree with Tyler? Explain why or why not.

Is the graph of \(y=\sin \left(2 x-\frac{\pi}{4}\right)\) the graph of \(y=\sin 2 x\) moved \(\frac{\pi}{4}\) units to the right? Explain why or why not.

In the interval \(0 \leq x \leq \pi,\) cos \(x\) decreases. Describe the change in sec \(x\) in the same interval.