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Chapter 5: Problem 42
What angle does the line \(y=4 x\) in the \(x y\) -plane make with the positive \(x\) -axis?
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Evaluate the indicated expressions assuming that $$ \begin{array}{ll} \cos x=\frac{1}{3} & \text { and } \sin y=\frac{1}{4} \\ \sin u=\frac{2}{3} & \text { and } \cos v=\frac{1}{5} \end{array} $$ Assume also that \(x\) and \(u\) are in the interval \(\left(0, \frac{\pi}{2}\right),\) that \(y\) is in the interval \(\left(\frac{\pi}{2}, \pi\right),\) and that \(v\) is in the interval \(\left(-\frac{\pi}{2}, 0\right) .\) $$\sin (u-v)$$
Suppose \(\theta\) is an angle such that \(\cos \theta\) is rational. Explain why \(\cos (2 \theta)\) is rational.
Evaluate the given quantities assuming that \(u\) and \(v\) are both in the interval \(\left(-\frac{\pi}{2}, 0\right)\) and \(\tan u=-\frac{1}{7} \quad\) and \(\quad \tan v=-\frac{1}{8}\) $$\cos v$$
Show that $$\sin ^{2}(2 \theta)=4\left(\sin ^{2} \theta-\sin ^{4} \theta\right)$$ for all \(\theta\).
Evaluate the indicated expressions assuming that $$ \begin{array}{ll} \cos x=\frac{1}{3} & \text { and } \sin y=\frac{1}{4} \\ \sin u=\frac{2}{3} & \text { and } \cos v=\frac{1}{5} \end{array} $$ Assume also that \(x\) and \(u\) are in the interval \(\left(0, \frac{\pi}{2}\right),\) that \(y\) is in the interval \(\left(\frac{\pi}{2}, \pi\right),\) and that \(v\) is in the interval \(\left(-\frac{\pi}{2}, 0\right) .\) $$\cos (x-y)$$
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