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Magazine Problem 108
Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. \(\operatorname{arcsec}\left(\sec \left(\frac{4 \pi}{3}\right)\right)\)
Problem 108
In Exercises \(82-128\), verify the identity. Assume that all quantities are defined. $$ \tan (\theta)+\cot (\theta)=\sec (\theta) \csc (\theta) $$
Problem 109
In Exercises \(82-128\), verify the identity. Assume that all quantities are defined. $$ \csc (\theta)-\sin (\theta)=\cot (\theta) \cos (\theta) $$
Problem 109
Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. \(\operatorname{arcsec}\left(\sec \left(\frac{5 \pi}{6}\right)\right)\)
Problem 110
Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. \(\operatorname{arcsec}\left(\sec \left(-\frac{\pi}{2}\right)\right)\)
Problem 110
In Exercises \(82-128\), verify the identity. Assume that all quantities are defined. $$ \cos (\theta)-\sec (\theta)=-\tan (\theta) \sin (\theta) $$
Problem 111
Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. \(\operatorname{arcsec}\left(\sec \left(\frac{5 \pi}{3}\right)\right)\)
Problem 112
Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. \(\operatorname{arccsc}\left(\csc \left(\frac{\pi}{6}\right)\right)\)
Problem 113
In Exercises \(82-128\), verify the identity. Assume that all quantities are defined. $$ \frac{1}{1-\cos (\theta)}+\frac{1}{1+\cos (\theta)}=2 \csc ^{2}(\theta) $$
Problem 113
Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. \(\operatorname{arcsc}\left(\csc \left(\frac{5 \pi}{4}\right)\right)\)
