Chapter 7: Analytic Trigonometry

Show that$\cos ({\sin }^{-1}\left(v\right)+{\cos }^{-1}(v\left)\right)=0$

Show that the functions f and g are identically equal.

$f\left(\theta \right)=\tan \theta +sec\theta ,g\left(\theta \right)=\frac{\cos \theta }{1-\sin \theta }$

Blood Pressure Blood pressure is a way of measuring the amount of force exerted on the walls of blood vessels. It is measured using two numbers: systolic (as the heart beats) blood pressure and diastolic (as the heart rests) blood pressure. Blood pressures vary substantially from person to person, but a typical blood pressure is 120/80, which means the systolic blood pressure is 120 mmHg and the diastolic blood pressure is 80 mmHg. Assuming that a person’s heart beats 70 times per minute, the blood pressureP of an individual after t seconds can be modeled by the function$P\left(t\right)=100+20\sin \left(\frac{7\mathrm{\pi }}{3}t\right)$

(a) In the interval [0, 1], determine the times at which the

blood pressure is 100 mmHg.

(b) In the interval [0, 1], determine the times at which the

blood pressure is 120 mmHg.

(c) In the interval[0, 1], determine the times at which the

blood pressure is between 100 and 105 mmHg.

Find the value of the number C:

$\frac{1}{2}{\sin }^{2}x+C=-\frac{1}{4}\cos \left(2x\right)$

Show$\sqrt{16+16{\tan }^2\theta }=4sec\theta if\frac{-\mathrm{\pi }}{2}<\theta <\frac{\mathrm{\pi }}{2}$

Show that the difference quotient for

role="math" localid="1646430733990" $$\begin{gathered} \frac{f(x+h)-f\left(x\right)}{h}=\frac{\sin (x+h)-\sin \left(x\right)}{h} \\ =\cos x·\frac{\sin h}{h}-\sin x·\frac{1-\cos h}{h} \end{gathered}$$

Show$\sqrt{9se{c}^2\theta -9}=3\tan \theta if\mathrm{\pi }\le \mathrm{\theta }<\frac{3\mathrm{\pi }}{2}$

The Ferris Wheel In 1893, George Ferris engineered the Ferris Wheel. It was 250 feet in diameter. If the wheel make 1revolutionevery40seconds, then the function $h\left(t\right)=125\sin \left(0.157t-\frac{\mathrm{\pi }}{2}\right)+125$ represents the height h, in feet, of a seat on the wheel as a function of time t, where t is measured in seconds. The ride begins when t = 0.

(a) During the first 40 seconds of the ride, at what time t is

an individual on the Ferris Wheel exactly 125 feet above

the ground?

(b) During the first 80 seconds of the ride, at what time t is

an individual on the Ferris Wheel exactly 250 feet above

the ground?

(c) During the first 40 seconds of the ride, over what interval

of timet is an individual on the Ferris Wheel more than

125 feet above the ground?

Show that the difference quotient for $f\left(x\right)=\cos x$is given by

role="math" localid="1646431168099" $$\begin{gathered} \frac{f(x+h)-f\left(x\right)}{h}=\frac{\cos (x+h)-\cos \left(x\right)}{h} \\ =-\sin x·\frac{\sin h}{h}-\cos x·\frac{1-\cos h}{h} \end{gathered}$$

Find the value of the number C:

$\frac{1}{2}{\cos }^{2}x+C=\frac{1}{4}\cos \left(2x\right)$