91Ó°ÊÓ

If \(\theta\) is an acute angle and \(\tan \theta=x,\) express the remaining five trigonometric functions in terms of \(x\).

A denotes the area of the sector of a circle of radius r formed by the central angle \(\theta .\) Find the missing quantity. Round answers to three decimal places. \(r=5\) miles, \(A=3\) square miles, \(\theta=?\)

Find the exact value of each of the remaining trigonometric functions of \(\theta\). $$ \cos \theta=-\frac{1}{3}, \quad 180^{\circ}<\theta<270^{\circ} $$

Several research papers use a sinusoidal graph to model blood pressure. Suppose an individual's blood pressure is modeled by the function $$ P(t)=20 \sin \left(\frac{7 \pi t}{3}\right)+100 $$ where the maximum value of \(P\) is the systolic pressure, which is the pressure when the heart contracts (beats), the minimum value is the diastolic pressure, and \(t\) is time, in seconds. The heart rate is the number of beats per minute. (a) What is the individual's systolic pressure? (b) What is the individual's diastolic pressure? (c) What is the individual's heart rate?

Movement of a Minute Hand The minute hand of a clock is 6 inches long. How far does the tip of the minute hand move in 15 minutes? How far does it move in 25 minutes? Round answers to two decimal places.

If \(\theta\) is an acute angle, explain why \(0<\sin \theta<1\).

Movement of a Pendulum A pendulum swings through an angle of \(20^{\circ}\) each second. If the pendulum is 40 inches long. how far does its tip move each second? Round answers to two decimal places.

If \(\theta\) is an acute angle, explain why sec \(\theta>1\).

The function below models the number of hours of daylight in Miami, Florida. $$ D(x)=1.615 \sin \left(\frac{2 \pi}{365} x-1.39\right)+12.135 $$ where \(x\) is the day of the year. (a) How many hours of daylight are there on the longest day? (b) How many hours of daylight are there on the shortest day? (c) What is the time between the longest and shortest days?

Area of a Sector Find the area of the sector of a circle of radius 3 centimeters formed by an angle of \(60^{\circ} .\) Round the answer to two decimal places.