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Chapter 4: Problem 6
Find the points where the line through the origin with slope 4 intersects the unit circle.
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In doing several of the exercises in this section, you should have noticed a relationship between \(\cos u\) and \(\sin v,\) along with a relationship between \(\sin u\) and \(\cos v\). What are these relationships? Explain why they hold.
Assume the surface of the earth is a sphere with diameter 7926 miles. Approximately how far does a ship travel when sailing along the equator in the Pacific Ocean from longitude \(170^{\circ}\) west to longitude \(120^{\circ}\) west?
In Exercises 5-38, find exact expressions for the indicated quantities, given that $$\cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2}$$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived in Examples 3 and 4 in Section 5.5.] \(\sin \left(-\frac{\pi}{12}\right)\)
(a) Show that $$x^{3}+x^{2} y+x y^{2}+y^{3}=\left(x^{2}+y^{2}\right)(x+y)$$ for all numbers \(x\) and \(y\). (b) Show that $$ \begin{aligned} \cos ^{3} \theta+\cos ^{2} \theta \sin \theta &+\cos \theta \sin ^{2} \theta+\sin ^{3} \theta \\ &=\cos \theta+\sin \theta \end{aligned} $$
Show that $$\cos (\pi-\theta)=-\cos \theta$$ for every angle \(\theta\).
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