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Magazine Problem 23
An initial amplitude \(k,\) damping constant \(c,\) and frequency \(f\) or period \(p\) are given. (Recall that frequency and period are related by the equation \(f=1 / p . )\) (a) Find a function that models the damped harmonic motion. Use a function of the form \(y=k e^{-c t} \cos \omega t\) in Exercises \(19-22,\) and of the form \(y=k e^{-c t} \sin \omega t\) in Exercises \(23-26\) (b) Graph the function. $$ k=7, \quad c=10, \quad p=\pi / 6 $$
Problem 24
\(23-44=\) Find the exact value of the expression, if it is defined. \(\cos \left(\cos ^{-1} \frac{2}{3}\right)\)
Problem 24
An initial amplitude \(k\), damping constant \(c,\) and frequency \(f\) or period \(p\) are given. (Recall that frequency and period are related by the equation \(f=1 / p .)\) (a) Find a function that models the damped harmonic motion. Use a function of the form \(y=k e^{-c t} \cos \omega t\) in Exercises \(19-22,\) and of the form \(y=k e^{-c t} \sin \omega t\) in Exercises \(23-26\) (b) Graph the function. $$ k=1, \quad c=1, \quad p=1 $$
Problem 24
Find the exact value of the trigonometric function at the given real number. (a) \(\sin \frac{25 \pi}{2} \quad\) (b) \(\cos \frac{25 \pi}{2} \quad\) (c) \(\cot \frac{25 \pi}{2}\)
Problem 24
\(17-28\) . Find the amplitude and period of the function, and sketch its graph. $$ y=4 \sin (-2 x) $$
Problem 24
23-32 \(\approx\) Find the terminal point \(P(X, y)\) on the unit circle determined by the given value of \(t\) $$ t=\frac{3 \pi}{2} $$
Problem 25
23-32 \(\approx\) Find the terminal point \(P(X, y)\) on the unit circle determined by the given value of \(t\) $$ t=\frac{5 \pi}{6} $$
Problem 25
Find the period and graph the function. $$ y=\frac{1}{2} \sec \left(x-\frac{\pi}{6}\right) $$
Problem 25
An initial amplitude \(k,\) damping constant \(c,\) and frequency \(f\) or period \(p\) are given. (Recall that frequency and period are related by the equation \(f=1 / p . )\) (a) Find a function that models the damped harmonic motion. Use a function of the form \(y=k e^{-c t} \cos \omega t\) in Exercises \(19-22,\) and of the form \(y=k e^{-c t} \sin \omega t\) in Exercises \(23-26\) (b) Graph the function. $$ k=0.3, \quad c=0.2, \quad f=20 $$
Problem 25
\(17-28\) . Find the amplitude and period of the function, and sketch its graph. $$ y=-2 \sin 2 \pi x $$
