91Ó°ÊÓ

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. \(\operatorname{arccsc}(-1)\)

Use the given information about \(\theta\) to find the exact values of \- \(\sin (2 \theta)\) \- \(\cos (2 \theta)\) \- \(\tan (2 \theta)\) \(\sin \left(\frac{\theta}{2}\right)\) - \(\cos \left(\frac{\theta}{2}\right)\) - \(\tan \left(\frac{\theta}{2}\right)\) $$ \sin (\theta)=\frac{5}{13} \text { where } \frac{\pi}{2}<\theta<\pi $$

Use the given information about \(\theta\) to find the exact values of \- \(\sin (2 \theta)\) \- \(\cos (2 \theta)\) \- \(\tan (2 \theta)\) \(\sin \left(\frac{\theta}{2}\right)\) - \(\cos \left(\frac{\theta}{2}\right)\) - \(\tan \left(\frac{\theta}{2}\right)\) $$ \sec (\theta)=\sqrt{5} \text { where } \frac{3 \pi}{2}<\theta<2 \pi $$

Find the exact value or state that it is undefined. $$ \sin \left(\arcsin \left(\frac{1}{2}\right)\right) $$

In Exercises \(43-58\), solve the equation, giving the exact solutions which lie in \([0,2 \pi)\). $$ \sin (6 x)+\sin (x)=0 $$

In Exercises \(43-57,\) find all of the angles which satisfy the equation. $$ \cot (\theta)=-1 $$

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

In Exercises \(58-65,\) solve the equation for \(t\). Give exact values. $$ \cot (t)=1 $$

Find the exact value or state that it is undefined. $$ \sin \left(\arcsin \left(-\frac{\sqrt{2}}{2}\right)\right) $$

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