91Ó°ÊÓ

Shortest Distance from Chicago to Honolulu Find the shortest distance from Chicago, latitude \(41^{\circ} 50^{\prime} \mathrm{N},\) longitude \(87^{\circ} 37^{\prime} \mathrm{W}\) to Honolulu, latitude \(21^{\circ} 18^{\prime} \mathrm{N},\) longitude \(157^{\circ} 50^{\prime} \mathrm{W}\). Round your answer to the nearest mile.

Establish each identity. $$(\tan \alpha+\tan \beta)(1-\cot \alpha \cot \beta)+(\cot \alpha+\cot \beta)(1-\tan \alpha \tan \beta)=0$$

The diameter of each wheel of a bicycle is 20 inches. If the wheels are turning at 336 revolutions per minute, how fast is the bicycle moving? Express the answer in miles per hour, rounded to the nearest integer.

Solve each equation on the interval \(0 \leq \theta<2 \pi\). $$ \sin \theta+\cos \theta=\sqrt{2} $$

Find the real zeros of each trigonometric function on the interval \(0 \leq \theta<2 \pi\) \(f(x)=\sin (2 x)-\sin x\)

If \(\cos \theta=\frac{24}{25},\) find the exact value of each of the remaining five trigonometric functions of acute angle \(\theta\)

Calculus Show that the difference quotient for \(f(x)=\sin x\) is given by $$ \begin{aligned} \frac{f(x+h)-f(x)}{h} &=\frac{\sin (x+h)-\sin x}{h} \\ &=\cos x \cdot \frac{\sin h}{h}-\sin x \cdot \frac{1-\cos h}{h} \end{aligned} $$

Area of an Octagon (a) The area \(A\) of a regular octagon is given by the formula \(A=8 r^{2} \tan \frac{\pi}{8}\) where \(r\) is the apothem, which is a line segment from the center of the octagon perpendicular to a side. See the figure. Find the exact area of a regular octagon whose apothem is 12 inches (b) The area \(A\) of a regular octagon is also given by the formula \(A=2 a^{2} \cot \frac{\pi}{8},\) where \(a\) is the length of a side. Find the exact area of a regular octagon whose side is 9 centimeters.

Area of a Dodecagon Part I A regular dodecagon is a polygon with 12 sides of equal length. See the figure. (a) The area \(A\) of a regular dodecagon is given by the formula \(A=12 r^{2} \tan \frac{\pi}{12},\) where \(r\) is the apothem, which is a line segment from the center of the polygon that is perpendicular to a side. Find the exact area of a regular dodecagon whose apothem is 10 inches. (b) The area \(A\) of a regular dodecagon is also given by the formula \(A=3 a^{2} \cot \frac{\pi}{12},\) where \(a\) is the length of a side of the polygon. Find the exact area of a regular dodecagon if the length of a side is 15 centimeters.

Geometry: Angle between Two Lines Let \(L_{1}\) and \(L_{2}\) denote two nonvertical intersecting lines, and let \(\theta\) denote the acute angle between \(L_{1}\) and \(L_{2}\) (see the figure). Show that $$ \tan \theta=\frac{m_{2}-m_{1}}{1+m_{1} m_{2}} $$ where \(m_{1}\) and \(m_{2}\) are the slopes of \(L_{1}\) and \(L_{2},\) respectively. [Hint: Use the facts that \(\tan \theta_{1}=m_{1}\) and \(\left.\tan \theta_{2}=m_{2} .\right]\)