1.
[latex]y = -3+2\sin x[/latex]
Chapter 7: Circular Functions
Suggested Homework Problems
For Problems 1–8, state the amplitude, period, and midline of the graph.
[latex]y = -3+2\sin x[/latex]
[latex]y = 4-3\cos x[/latex]
[latex]y = -\cos 4x[/latex]
[latex]y = -\sin 3x[/latex]
[latex]y = -5\sin \dfrac{x}{3}[/latex]
[latex]y = 6\cos \dfrac{x}{2}[/latex]
[latex]y = 1-\cos \pi x[/latex]
[latex]y = 2+\sin 2\pi x[/latex]
In Problems 9–16, we use transformations to sketch graphs of the functions in Problems 1–8. Sketch one cycle of each graph by hand and label scales on the axes.
For Problems 17–22, write an equation for the graph using sine or cosine.
For Problems 23–30,
In Problems 31–36, we use a table of values to sketch circular functions.
[latex]y=2-5\cos 2t[/latex]
| [latex]t[/latex] | [latex]2t[/latex] | [latex]\cos 2t[/latex] | [latex]-5\cos 2t[/latex] | [latex]2-5\cos 2t[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]0[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\pi[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]2\pi[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
[latex]y=-2+4\sin 3t[/latex]
| [latex]t[/latex] | [latex]3t[/latex] | [latex]\sin 3t[/latex] | [latex]4\sin 3t[/latex] | [latex]-2+4\sin 3t[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]0[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\pi[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]2\pi[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
[latex]y=1+3\cos \dfrac{t}{2}[/latex]
| [latex]t[/latex] | [latex]\dfrac{t}{2}[/latex] | [latex]\cos \dfrac{t}{2}[/latex] | [latex]3\cos \dfrac{t}{2}[/latex] | [latex]1+3\cos \dfrac{t}{2}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]0[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\pi[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]2\pi[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
[latex]y=-2-3\sin \dfrac{t}{4}[/latex]
| [latex]t[/latex] | [latex]\dfrac{t}{4}[/latex] | [latex]\sin \dfrac{t}{4}[/latex] | [latex]3\sin \dfrac{t}{4}[/latex] | [latex]-2-3\sin \dfrac{t}{4}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]0[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\dfrac{\pi}{2}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\pi[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\dfrac{3\pi}{2}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]2\pi[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
[latex]y=-3+2\sin \dfrac{t}{3}[/latex]
| [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
[latex]y=-1+4\cos \dfrac{t}{6}[/latex]
| [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
| [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] | [latex]\hphantom{0000}[/latex] |
For Problems 37–44, label the scales on the axes for the graph.
[latex]y=3-4\sin 2x[/latex]
[latex]y=2\cos 5x+2[/latex]
[latex]y=\dfrac{1}{2} \sin 3x+\dfrac{3}{2}[/latex]
[latex]y = \dfrac{2}{5}\cos 6x +\dfrac{4}{5}[/latex]
[latex]50-30 \sin \dfrac{x}{4}[/latex]
[latex]25 \cos \dfrac{x}{3} + 15[/latex]
[latex]y = 4 \sin \pi x - 3[/latex]
[latex]y = \dfrac{1}{2} \cos \dfrac{\pi x}{2} + 2[/latex]
The height of the tide in Cabot Cove can be approximated by a sinusoidal function. At 5 am on July 23, the water level reached its high mark at the 20-foot line on the pier, and at 11 am, the water level was at its lowest at the 4-foot line.
The population of mosquitoes at Marsh Lake is a sinusoidal function of time. The population peaks around June 1, at about 6000 mosquitoes per square kilometer, and is smallest on December 1, at 1000 mosquitoes per square kilometer.
The paddlewheel on the Delta Queen steamboat is 28 feet in diameter and is rotating once every ten seconds. The bottom of the paddlewheel is 4 feet below the surface of the water.
Delbert’s bicycle wheel is 24 inches in diameter, and he has a light attached to the spokes 10 inches from the center of the wheel. It is dark, and he is cycling home slowly from work. The bicycle wheel makes one revolution every second.
For Problems 49–52, write an equation for the sinusoidal function whose graph is shown.
The number of hours of daylight in Salt Lake City varies from a minimum of 9.6 hours on the winter solstice to a maximum of 14.4 hours on the summer solstice. Time is measured in months, starting at the winter solstice.
A weight is 6.5 feet above the floor, suspended from the ceiling by a spring. The weight is pulled down to 5 feet above the floor and released, rising past 6.5 feet in 0.5 seconds before attaining its maximum height of feet. The weight oscillates between its minimum and maximum height.
The voltage used in U.S. electrical current changes from 155V to 155V and back 60 times each second.
Although the moon is spherical, what we see from earth looks like a disk, sometimes only partly visible. The percentage of the moon’s disk that is visible varies between 0 (at new moon) and 100 (at full moon) over a 28-day cycle.
For Problems 53–58,
[latex]y=\tan 2x[/latex]
[latex]y=\tan 4x[/latex]
[latex]y=4+2\tan 3x[/latex]
[latex]y=3+\dfrac{1}{2} \tan 2x[/latex]
[latex]y=3-\tan \dfrac{x}{4}[/latex]
[latex]y=1-2\tan \dfrac{x}{3}[/latex]
For Problems 59–64, use the graph to find all solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex]
[latex]3\cos 4x = 1.5[/latex]
[latex]2\sin 3x = -\sqrt{2}[/latex]
[latex]2+3\sin 2x = 0.5[/latex]
[latex]2+4\cos 2x = 4[/latex]
[latex]-3+\tan 3x = -2[/latex]
[latex]2+\tan 4x = 3[/latex]
For Problems 65–70,