91Ó°ÊÓ

Is the cosecant function even, odd, or neither? Is its graph symmetric? With respect to what?

Nautical Miles A nautical mile equals the length of arc subtended by a central angle of 1 minute on a great circle† on the surface of Earth. See the figure. If the radius of Earth is taken as 3960 miles, express 1 nautical mile in terms of ordinary, or statute, miles.

From a parking lot, you

want to walk to a house on the beach. The house is located

1500 feet down a paved path that parallels the ocean, which

is 500 feet away. See the illustration. Along the path you can

walk 300 feet per minute, but in the sand on the beach you

can only walk 100 feet per minute.

The time T to get from the parking lot to the beach

house can be expressed as a function of the angle u shown in

the illustration and is

$T\left(\mathrm{θ}\right)=5-\frac{5}{3\tan \mathrm{θ}}+\frac{5}{\sin \mathrm{θ}},0<\mathrm{θ}<\frac{\mathrm{π}}{2}$

Calculate the time T if you walk directly from the parking lot

to the house.

Two oceanfront homes are

located 8 miles apart on a straight stretch of beach, each a

distance of 1 mile from a paved path that parallels the ocean.

Sally can jog 8 miles per hour on the paved path, but only 3

miles per hour in the sand on the beach. Because a river flows

directly between the two houses, it is necessary to jog in the

sand to the road, continue on the path, and then jog directly

back in the sand to get from one house to the other. See the

illustration. The time T to get from one house to the other as

a function of the angle $\mathrm{θ}$shown in the illustration is

$T\left(\mathrm{θ}\right)=1+\frac{2}{3Sin\mathrm{θ}}-\frac{1}{4\tan \mathrm{θ}}0<\mathrm{θ}<\frac{\mathrm{π}}{2}$

(a) Calculate the time T for tan$\mathrm{θ}=\frac{1}{4}$

(b) Describe the path taken.

(c) Explain why $\mathrm{θ}$must be larger than 14°.

Projectile Motion - An object is propelled upward at an angle $\mathrm{θ}$between $45°$and $90°$to the horizontal with an initial

velocity of ${\mathrm{ν}}_o$feet per second from the base of an inclined plane that makes an angle of 45° with the horizontal. See the

illustration. If air resistance is ignored, the distance R that it travels up the inclined plane as a function of $\mathrm{θ}$is given by

$R\left(\mathrm{θ}\right)=\frac{{{\mathrm{ν}}_o}^2\sqrt{2}}{32}\left[\sin \left(2\mathrm{θ}\right)-\cos \left(2\mathrm{θ}\right)-1\right]$

(a) Find the distance R that the object travels along the inclined plane if the initial velocity is 32 feet per second and $\mathrm{θ}=60°$

(b) Graph R = R$\left(\mathrm{θ}\right)$if the initial velocity is 32 feet per second.

(c) What value of $\mathrm{θ}$makes R largest?

Write a brief paragraph that explains how to quickly compute the trigonometric functions of 30°, 45°, and 60°.

In Problems 16–23, find the exact value of each of the remaining trigonometric functions.

$cot\mathrm{θ}=-2,\frac{\mathrm{π}}{2}<\mathrm{θ}<\mathrm{π}$.

Logan has a garden in the shape of a sector of a circle; the outer rim of the garden is $25$feet long and the central angle of the sector is $50°$. She wants to add a $3$-foot wide walk to the outer rim; how many square feet of paving blocks will she need to build the walk?

Hurricanes are categorized using the Saffir- Simpson Hurricane Scale, with winds $111-130$miles per hour (mph) corresponding to a category $3$ hurricane, winds $131-155$mph corresponding to a category $4$ hurricane, and winds in excess of $155$ mph corresponding to a category
$5$hurricane. The following data represent the number of major hurricanes in the Atlantic Basin (category $3,4$or $5$) each decade from $1921to2010$.

(a) Draw a scatter diagram of the data for one period.

(b) Find a sinusoidal function of the form $y=A\sin (\mathrm{Ó\neg }x-\mathrm{Ï•})+B$that models the data.

(c) Draw the sinusoidal function found in part (b) on the scatter diagram.

(d) Use a graphing utility to find the sinusoidal function of best fit.

(e) Draw the sinusoidal function of best fit on a scatter diagram of the data.

An angle $\mathrm{θ}$is in ________ _________ if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with the positive x-axis.