91Ó°ÊÓ

Derive the cofunction identity \(\sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta\) using the difference formula for sine.

\(g(x)=\sec 2 x\)

\(g(x)=\csc 2 x\)

Your friend claims it is possible to use the difference formula for tangent to derive the cofunction identity \(\tan \left(\frac{\pi}{2}-\theta\right)=\cot \theta\). Is your friend correct? Explain your reasoning.

Graph the function. \(g(x)=\cos \frac{1}{2}(x-3 \pi)-5\)

Describe and correct the error in finding the period of the function \(y=\sin \frac{2}{3} x\).

WRITING Explain why there is more than one tangent function whose graph passes through the origin and has asymptotes at \(x=-\pi\) and \(x=\pi\).

When a wave travels through a taut string, the displacement \(y\) of each point on the string depends on the time \(t\) and the point's position \(x\). The equation of a standing wave can be obtained by adding the displacements of two waves traveling in opposite directions. Suppose a standing wave can be modeled by the formula \(y=A \cos \left(\frac{2 \pi t}{3}-\frac{2 \pi x}{5}\right)+A \cos \left(\frac{2 \pi t}{3}+\frac{2 \pi x}{5}\right)\). When \(t=1\), show that the formula can be rewritten as \(y=-A \cos \frac{2 \pi x}{5}\).

Describe and correct the error in determining the point where the maximum value of the function \(y=2 \sin \left(x-\frac{\pi}{2}\right)\) occurs.

USING EQUATIONS Graph one period of each function. Describe the transformation of the graph of its parent function. a. \(g(x)=\sec x+3\) b. \(g(x)=\csc x-2\) c. \(g(x)=\cot (x-\pi)\) d. \(g(x)=-\tan x\)