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Magazine Problem 6
For \(\theta=-80^{\circ}\), evaluate each of the following: (a) \(\cos \frac{\theta}{2}\) (b) \(\frac{\cos \theta}{2}\)
Problem 6
Find exact expressions for th indicated quantities. The following information will be useful:\(\cos 22.5^{\circ}=\frac{\sqrt{2+\sqrt{2}}}{2}\) and \(\sin 22.5^{\circ}=\frac{\sqrt{2-\sqrt{2}}}{2}\) ; \(\cos 18^{\circ}=\sqrt{\frac{\sqrt{5}+5}{8}}\) and \(\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}\). [The value for \(\sin 22.5^{\circ}\) used here was derived in Example 5 in Section 6.3; the other values were derived in Exercise 64 and Problems 101 and 102 in Section 6.3.] $$ \begin{array}{l} \cos 48^{\circ} \\ \text { [Hint: } 48=30+18] \end{array} $$
Problem 6
Convert the polar coordinates given for each point to rectangular coordinates in the \(x y\) -plane. $$ r=7, \theta=\frac{\pi}{4} $$
Problem 6
Suppose \(\mathbf{u}=(-3,2)\) and \(\mathbf{v}=(-2,-1)\) (a) Draw a figure illustrating the sum of \(\mathbf{u}\) and \(\mathbf{V}\) as arrows. (b) Compute the sum \(\mathbf{u}+\mathbf{v}\) using coordinates.
Problem 7
8 For \(\theta=65^{*},\) evaluate each of the following: (a) \(\sin \frac{\theta}{2}\) (b) \(\frac{\sin \theta}{2}\)
Problem 7
Find exact expressions for th indicated quantities. The following information will be useful:\(\cos 22.5^{\circ}=\frac{\sqrt{2+\sqrt{2}}}{2}\) and \(\sin 22.5^{\circ}=\frac{\sqrt{2-\sqrt{2}}}{2}\) ; \(\cos 18^{\circ}=\sqrt{\frac{\sqrt{5}+5}{8}}\) and \(\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}\). [The value for \(\sin 22.5^{\circ}\) used here was derived in Example 5 in Section 6.3; the other values were derived in Exercise 64 and Problems 101 and 102 in Section 6.3.] $$ \sin 82.5^{\circ} $$
Problem 7
Convert the polar coordinates given for each point to rectangular coordinates in the \(x y\) -plane. $$ r=8, \theta=\frac{\pi}{3} $$
Problem 7
Suppose \(\mathbf{u}=(2,1)\) and \(\mathbf{v}=(3,1)\) (a) Draw a figure using arrows illustrating the difference \(\mathbf{u}-\mathbf{v}\) (b) Compute the difference \(\mathbf{u}-\mathbf{v}\) using coordinates.
Problem 8
For \(\theta=9\) radians, evaluate each of the following: (a) \(\sin \frac{\theta}{2}\) (b) \(\frac{\sin \theta}{2}\)
Problem 8
Suppose \(\mathbf{u}=(-3,2)\) and \(\mathbf{v}=(-2,-1)\) (a) Draw a figure using arrows illustrating the difference \(\mathbf{u}-\mathbf{v}\). (b) Compute the difference \(\mathbf{u}-\mathbf{v}\) using coordinates.
