E-RA03 · Can two proposed ratios belong to one angle?
For a real constant \(k\), an angle \(\theta\in[0,2\pi)\) is to satisfy \(\displaystyle \cos\theta=\frac{k-1}{4}\) and \(\displaystyle \sin\theta=\frac{k+1}{4}\).
Find all possible values of \(k\), checking that the resulting ratios are valid.
Identify the quadrant of \(\theta\) for each value of \(k\).
Find the exact tangent in each case. Express the corresponding principal value \(\tan^{-1}(\tan\theta)\) in terms of \(\theta\).
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