91Ó°ÊÓ

Graph the functions. $$ y=1-\left[\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right]^{2} $$

A bicycle ramp is made so that it can easily be raised and lowered for different levels of competition. For the advance division, the angle formed by the ramp and the ground is \(\theta\) such that \(\sin \theta=\frac{2 \sqrt{2}}{3}\). For the novice division, the angle \(\theta\) is cut in half to lower the ramp. What is \(\tan \left(\frac{\theta}{2}\right)\), the steepness of the ramp?

Determine whether each equation is an identity, a conditional equation, or a contradiction. $$ \sec ^{2} x-\tan ^{2} x=1 $$

Determine whether each equation is an identity, a conditional equation, or a contradiction. $$ \sqrt{\sin ^{2} x+\cos ^{2} x}=1 $$

An electric field \(E\) of a wave with constant amplitude \(A\) propagating a distance \(z\) is given by $$ E=A \cos (k z-c t) $$ where \(k\) is the propagation wave number, which is related to the wavelength \(\lambda\) by \(k=\frac{2 \pi}{\lambda}, c=3.0 \times 10^{8}\) meters per second is the speed of light in a vacuum, and \(t\) is time in seconds. Use the cosine difference identity to express the electric field in terms of both sine and cosine functions. When the quotient of the propagation distance \(z\) and the wavelength \(\lambda\) are equal to an integer, what do you notice?

Determine whether each statement is true or false. If an equation has an infinite number of solutions, then it is an identity.

Find the exact value of \(\tan \left(-\frac{7 \pi}{6}\right)\). Solution: Evaluate the tangent functions on the right. $$ =\frac{0+\frac{1}{\sqrt{3}}}{1-0} $$ Simplify. $$ =\frac{\sqrt{3}}{3} $$ This is incorrect. What mistake was made?

Determine whether the statement is true or false. The following is an identity true for all values in the domain of the functions: \(\tan ^{2} x-\sec ^{2} x=1\).

In what quadrants is the equation \(-\cos \theta=\sqrt{1-\sin ^{2} \theta}\) true?

Verify that \(\sin (A+B+C)=\sin A \cos B \cos C+\) \(\cos A \sin B \cos C+\cos A \cos B \sin C-\sin A \sin B \sin C\).