Shown in the figure is the area served by a communications satellite circling a planet of radius $R$ at an altitude $a$. The portion of the planet's surface within range of the satellite is a spherical cap of depth $d$ and surface area $A=2 \pi R d$.

Express $d$ in terms of $R$ and $\theta$.

Verify the identity by transforming the lefthand side into the right-hand side.

$$\cot \theta \sec \theta=\csc \theta$$

Sketch the graph of the equation.

$$y=\sec \left(2 x-\frac{\pi}{2}\right)$$

Find an equation using the cosecant function that has the same graph as $y=\sec x$.

Sketch the graph of the equation.

$$y=\sec \left(2 x-\frac{\pi}{2}\right)$$

Refer to the graph of the equation on the specified interval. Find all values of $x$ such that for the real number $a$, $y<a$,

$$y=\cos x ; \quad[-2 \pi, 2 \pi] ; \quad a=-\frac{1}{2}$$

Communications satellite Shown in the left part of the figure is a communications satellite with an equatorial orbitthat is, a nearly circular orbit in the plane determined by Earth's equator. If the satellite circles Earth at an altitude of $a=22,300 \mathrm{mi}$, its speed is the same as the rotational speed of Earth; to an observer on the equator, the satellite appears to be stationary - that is, its orbit is synchronous.

As shown in the right part of the figure, three satellites are equally spaced in equatorial synchronous orbits.

Refer to the graph of the equation on the specified interval. Find all values of $x$ such that for the real number $a$, $y>a$,

$$y=\cos x ; \quad[-2 \pi, 2 \pi] ; \quad a=-\frac{1}{2}$$

Communications satellite Shown in the left part of the figure is a communications satellite with an equatorial orbitthat is, a nearly circular orbit in the plane determined by Earth's equator. If the satellite circles Earth at an altitude of $a=22,300 \mathrm{mi}$, its speed is the same as the rotational speed of Earth; to an observer on the equator, the satellite appears to be stationary - that is, its orbit is synchronous.

Using $R=4000 \mathrm{mi}$ for the radius of Earth, determine the percentage of the equator that is within signal range of such a satellite.

Refer to the graph of the equation on the specified interval. Find all values of $x$ such that for the real number $a$, $y=a$,

$$y=\cos x ; \quad[-2 \pi, 2 \pi] ; \quad a=-\frac{1}{2}$$

Sketch the graph of the equation.

$$y=\sec \left(\frac{1}{2} x+\pi\right)$$

An expert cyclist can attain a speed of $40 \mathrm{mi} / \mathrm{hr}$. If the sprocket assembly has $r_1=5$ in., $r_2=2$ in., and the wheel has a diameter of $28$ inches, approximately how many revolutions per minute of the front sprocket wheel will produce a speed of $40 \mathrm{mi} / \mathrm{hr}$ ? (Hint: First change $40 \mathrm{mi} / \mathrm{hr}$ to in./sec.)

Verify the identity by transforming the lefthand side into the right-hand side.

$$\frac{\csc \theta}{\sec \theta}=\cot \theta \quad$$

Find an equation using the cotangent function that has the same graph as $y=\tan x$.