91Ó°ÊÓ

In Exercises \(43-58\), solve the equation, giving the exact solutions which lie in \([0,2 \pi)\). $$ \sqrt{2} \cos (x)-\sqrt{2} \sin (x)=1 $$

Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) and that the range of arccosecant is \(\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\) when finding the exact value. \(\operatorname{arcsec}(-2)\)

Graph the function with the help of your calculator and discuss the given questions with your classmates. \(f(x)=e^{-0.1 x}(\cos (2 x)+\sin (2 x))\). Graph \(y=\pm e^{-0.1 x}\) on the same set of axes and describe the behavior of \(f\).

Sketch the oriented arc on the Unit Circle which corresponds to the given real number. $$ t=12 $$

In Exercises \(49-58\), use the given information about \(\theta\) to find the exact values of \- \(\sin (2 \theta)\) \- \(\cos (2 \theta)\) \- \(\tan (2 \theta)\) \(\sin \left(\frac{\theta}{2}\right)\) - \(\cos \left(\frac{\theta}{2}\right)\) - \(\tan \left(\frac{\theta}{2}\right)\) $$ \sin (\theta)=-\frac{7}{25} \text { where } \frac{3 \pi}{2}<\theta<2 \pi $$

Use your calculator to approximate the given value to three decimal places. $$ \cos (-2.01) $$

In Exercises \(43-57,\) find all of the angles which satisfy the equation. $$ \cot (\theta)=0 $$

Use the given information about \(\theta\) to find the exact values of \- \(\sin (2 \theta)\) \- \(\cos (2 \theta)\) \- \(\tan (2 \theta)\) \(\sin \left(\frac{\theta}{2}\right)\) - \(\cos \left(\frac{\theta}{2}\right)\) - \(\tan \left(\frac{\theta}{2}\right)\) $$ \cos (\theta)=\frac{28}{53} \text { where } 0<\theta<\frac{\pi}{2} $$

A yo-yo which is 2.25 inches in diameter spins at a rate of 4500 revolutions per minute. How fast is the edge of the yo-yo spinning in miles per hour? Round your answer to two decimal places.

Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) and that the range of arccosecant is \(\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\) when finding the exact value. \(\operatorname{arcsec}(-\sqrt{2})\)