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Discovering Advanced Algebra An Investigative Approach 1st edition Jerald Murdock, Ellen Kamischke, Eric Kamischke - Solutions

Solve for Î¸. Express your answers in radians.a.b. c.
Suppose you are biking down a hill at 24 mi/h. What is the angular speed, in radians per second, of your 27-inch-diameter bicycle wheel?
A sector of a circle with radius 8 cm has central angle 4π/7. a. Find the area of the sector. b. Set up a proportion of the area of the sector to the total area of the circle, and a proportion of the central angle of the sector to the total central angle measure. c. Solve your proportion from 13b
Sitting at your desk, you are approximately 6350 km from the center of Earth. Consider your motion relative to the center of Earth, as Earth rotates on its axis. a. What is your angular speed? b. What is your speed in km/h? c. What is your speed in mi/h?
The two cities Minneapolis, Minnesota, and Lake Charles, Louisiana, lie on the 93° W longitudinal line. The latitude of Minneapolis is 45° N (45° north of the equator) whereas the latitude of Lake Charles is 30° N. The radius of Earth is approximately 3960 mi.a. Calculate the
List the transformations of each graph from its parent function.a. y = 2 + (x + 4)2b.c. y + 1 = |x - 3 | d. y = 3 - 2 |x + 1|
Write an equation for each graph.a.b. c. d.
Find a second value of θ that gives the same trigonometric value as the angle given. Use domain 0° ≤ θ ≤ 360°. a. sin 23° = sin θ b. sin 216° = sin θ c. cos 342° = cos θ d. cos 246° = cos θ
While Yolanda was parked at her back steps, a slug climbed onto the wheel of her go-cart just above where the wheel makes contact with the ground. When Yolanda hopped in and started to pull away, she did not notice the slug until her wheels had rotated 2 times. Yolanda's wheels have a 12 cm
Find the length of the intercepted arc for each central angle. a. r = 3 and θ = 2π/3 b. r = 1 and θ = 1 c. d = 5 and θ = π/6
Sketch a rectangle with length a and width 2a. Inscribe an ellipse in the rectangle. What is the eccentricity of the ellipse? Why?
Circles O and P, shown at right, are tangent at A. Explain why AB = BC.
A river flowing at 1.50 m/s runs over an 80 m high cliff and into a lake below.a. Write and graph parametric equations to model the water's path from the cliff to the lake.b. Use the trace feature to approximate the time it takes for the water to reach the surface of the lake. Then find the
Draw a large copy of this diagram on your paper. Each angle shown has a reference angle of 0°, 30°, 45°, 60°, or 90°.a. Find the counterclockwise degree rotation of each segment from the positive x-axis. Write your answers in both degrees and radians. b. Find the exact
The minute hand of a clock is 15.2 cm long. a. What is the distance the tip of the minute hand travels during 40 minutes? b. At what speed is the tip moving, in cm/min? c. What is the angular speed of the tip in radians/minute?
On your paper, graph y = sin x over the domain 0 ≤ x ≤ 2. a. On the x-axis, label all the x-values that are multiples of π/6. b. On the x-axis, label all the x-values that are multiples of π/4. c. What x-values in this domain correspond to a maximum value of sin x? A minimum value of sin x,
On your paper, graph y = cos x over the domain 0 ≤ x ≤ 2π. a. On the x-axis, label all the x-values that are multiples of π/6. b. On the x-axis, label all the x-values that are multiples of π/4. c. What x-values in this domain correspond to a maximum value of cos x? A minimum value of cos x?
For 1a-f, write an equation for each sinusoid as a transformation of the graph of either y = sin x or y = cos x. More than one answer is possible.
Graph y = tan x on your calculator. Use Radian mode with - 2π ≤ x ≤ 2π and - 5 ≤ y ≤ 5. a. What happens at x = π/2? Explain why this is so, and name other values when this occurs. b. What is the period of y = tan x? c. Explain, in terms of the definition of tangent, why the values of tan
Write equations for sinusoids with these characteristics: a. a cosine function with amplitude 1.5, period π, and phase shift -π/2 b. a sine function with minimum value - 5, maximum value -1 and one cycle starting at x = π/4 and ending at x = 3π/4 c. a cosine function with period 6π, phase
The table gives the number of hours between sunrise and sunset for the period between December 21, 1995, and July 29, 1996, in New Orleans, Louisiana, which is located at approximately 30° N latitude.a. Assign December 31 as day zero, let x represent the number of days after December 31, and
Find the measure of the labeled angle in each triangle.a.b.
Make a table of angle measures from 0° to 360° by 15° increments. Then find the radian measure of each angle. Express the radian measure as a multiple of π.
The second hand of a wristwatch is 0.5 cm long. a. What is the speed, in meters per hour, of the tip of the second hand? b. How long would the minute hand of the same watch have to be for its tip to have the same speed as the second hand? c. How long would the hour hand of the same watch have to be
Consider these three functions: a. Find the inverse of each function. i. f(x) = -3/2x + 6 ii. g(x) = (x+2)2 -4 iii. h(x) = 1.3 x+6 -8 b. Graph each function and its inverse. c. Which of the inverses, if any, are functions?
Use what you know about the unit circle to find possible values of in each equation. Use domain 0° ≤ θ ≤ 360° or 0 ≤ θ ≤ 2π. a. cos θ = sin 86o b. sin θ = cos 19π/12 c. sin θ = cos 123o d. cos θ = sin 7π/6
For 2a-f, write an equation for each sinusoid as a transformation of the graph of either y = sin x or y = cos x. More than one answer is possible
Consider the graph of y = k + b sin (x-h)/h . a. What effect does k have on the graph of y = k + sin x? b. What effect does b have on the graph of y = b sin x? What is the effect if b is negative? c. What effect does a have on the graph of y = sin(y/a)? d. What effect does h have on the graph of y
Sketch the graph of y = 2sin(x/3) - 4. Use a calculator to check your sketch.
Write three different equations for the graph at right.
The percentage of the lighted surface of the Moon that is visible from Earth can be modeled with a sinusoid. Assume that tonight the Moon is full (100%) and in 14 days it will be a new moon (0%). a. Define variables and find a sinusoidal function that models this situation. b. What percentage will
You may have noticed that a fluorescent light flickers, especially when it is about to blow out. Fluorescent lights do not produce constant illumination like incandescent lights. Ideally, fluorescent light cycles sinusoid ally from dim to bright 60 times per second. At a certain distance from the
Imagine a unit circle in which a point is rotated A radians counterclockwise about the origin from the positive x-axis. Copy this table and record the x-coordinate and y-coordinate for each angle. Then use the definition of tangent to find the slope of the segment that connects the origin to each
Find the principal value of each expression to the nearest tenth of a degree and then to the nearest hundredth of a radian. a. sin -1 0.4665 b. sin -1 (- 0.2471) c. cos -1 (- 0.8113) d. cos -1 0.9805
In Î”ABC, AB = 7 cm, CA = 3.9 cm, and m ˆ B = 27°. Find the two possible measurements for ˆ C.
Consider the inverse of the tangent function. a. Find the values of tan -1 x for several positive and negative values of x. b. Based on these answers, predict what the graph of y = tan -1 x will look like. Sketch your prediction. c. How does the graph of y = tan -1x compare to the graph of x = tan
Shown at right are a constant function and two cycles of a cosine graph over the domain 0° ≤ x ≤ 720°. The first intersection point is shown. What are the next three intersection values?
Find the exact value of each expression.a.b. c. d.
When a beam of light passes through a polarized lens, its intensity is cut in half, or I1 = 0.5I0. To further reduce the intensity, you can place another polarized lens in front of it. The intensity of the beam after passing through the second lens depends on the angle of the second filter to the
Convert a. 7π/10 radians to degrees b. - 205° to radians c. 5π radians per hour to degrees per minute
Find all roots, real or nonreal. Give exact answers. a. 2x2 - 6x + 3 = 0 b. 13x - 2x2 = 6 c. 3x2 + 4x + 4 = 0
Find all four values of x between - 2π and 2 π that satisfy each equation. a. sinx = sinπ/6 b. cosx = cos3π/8 c. cosx = cos0.47 d. sinx = sin1.47
Illustrate the answers to Exercise 2 by plotting the points on the graph of the sine or cosine curve. In Exercise 2 a. sinx = sinπ/6 b. cosx = cos3π/8 c. cosx = cos0.47 d. sinx = sin1.47
Illustrate the answers to Exercise 2 by drawing segments on a graph of the unit circle. In Exercise 2 a. sinx = sinπ/6 b. cosx = cos3π/8 c. cosx = cos0.47 d. sinx = sin1.47
On the same coordinate axes, create graphs of y = 2 cos(x + π/4) and its inverse.
Find values of x that satisfy the conditions given.
How many solutions to the equation - 2.6 = 3 sin x occur in the first three positive cycles of the function y = 3 sin x? Explain your answer.
Find the measure of the largest angle of a triangle with sides 4.66 m, 5.93 m, and 8.54 m
Find the first four positive solutions. Give exact values in radians. a. cos x = 0.5 b. sin x = - 0.5
An AM radio transmitter generates a radio wave given by a function in the form f (t) = A sin 2000 nt. The variable n represents the location on the broadcast dial, 550 ≤ n ≤ 1600, and t is the time in seconds. a. For radio station WINS, located at 1010 on the AM radio dial, what is the period
The number of hours of daylight, y, on any day of the year in Philadelphia can be modeled using the equation y = 12 + 2.4 sin (2π(x-80)/365), where x represents the day number (with January 1 as day 1). a. Find the amount of sunlight in Philadelphia on day 354, the shortest day of the year (the
A popular amusement park ride is the double Ferris wheel. Each small wheel takes 20 s to make a single rotation. The two-wheel set takes 30 s to rotate once. The dimensions of the ride are given in the diagram.a. Sandra gets on at the foot of the bottom wheel. Write an equation that will model her
Use geometry software to simulate Sandra's ride on the double ferris wheel in Exercise 12. Describe your steps.
Solve tanθ = - 1.111 graphically. Use domain - 180° ≤ θ ≤ 360°. Round answers to the nearest degree
Which has the larger area, an equilateral triangle with side 5 cm or a sector of a circle with radius 5 cm and arc length 5 cm? Give the area of each shape to the nearest 0.1 cm.
Find the equation of the circle with center (- 2, 4) that has tangent line 2x - 3y - 6 = 0.
Consider the equation P(x) = 2x3 - x2 - 10x + 5. a. List all possible rational roots of P(x). b. Find any actual rational roots. c. Find exact values for all other roots. d. Write P(x) in factored form.
Circles C1, C2, . . . are tangent to the sides of P and to the adjacent circle(s). The radius of circle C1 is 6. The measure of P is 60°.a. What are the radii of C2 and C3?b. What is the radius of Cn?
Find all solutions for 0 ≤ x ≤ 2, rounded to the nearest thousandth. a. 2 sin (x + 1.2) - 4.22 = - 4 b. 7.4 cos (x - 0.8) + 12.3 = 16.4
Consider the graph of the function h = 5 + 7 sin (2π(t-9)/11). a. What are the vertical translation and average value? b. What are the vertical stretch factor, minimum and maximum values, and amplitude? c. What are the horizontal stretch factor and period? d. What are the horizontal translation
Consider the graph of the function h = 18 - 17 cos (2π(t+16)/15). a. What are the vertical translation and average value? b. What are the vertical stretch factor, minimum and maximum values, and amplitude? c. What are the horizontal stretch factor and period? d. What are the horizontal translation
A walker moves counterclockwise around a circle with center (1.5, 2) and radius 1.2 m and completes a cycle in 8 s. A recorder walks back and forth along the x-axis, staying even with the walker, with a motion sensor pointed toward the walker. What equation models the (time, distance) relationship?
A mass attached to a string is pulled down 3 cm from its resting position and then released. It makes ten complete bounces in 8 s. At what times during the first 2 s was the mass 1.5 cm above its resting position?
Household appliances are typically powered by electricity through wall outlets. The voltage provided varies sinusoidally between - 110 √2 volts and 110√2 volts, with a frequency of 60 cycles per second. a. Use a sine or cosine function to write an equation for (time, voltage). b. Sketch and
The time between high and low tide in a river harbor is approximately 7 h. The high-tide depth of 16 ft occurs at noon and the average harbor depth is 11 ft. a. Write an equation modeling this relationship. b. If a boat requires a harbor depth of at least 9 ft, find the next two time periods when
Two masses are suspended from springs, as shown. The first mass is pulled down 3 cm from its resting position and released. A second mass is pulled down 4 cm from its resting position. It is released just as the first mass passes its resting position on its way up. When released, each mass makes 12
Find the first three positive x-values that make each equation true. a. sec x = - 2.5 b. csc x = 0.4
Sketch a graph for each equation with domain 0 ≤ x < 4π. Include any asymptotes and state the x-values at which the asymptotes occur. a. y = csc x b. y = sec x c. y = cot x
Write an equation for each graph. More than one answer is possible. Use your calculator to check your work.a.b. c. d.
A pharmacist has 100 mL of a liquid medication that is 60% concentrated. This means that in 100 mL of the medication, 60 mL is pure medicine and 40 mL is water. She alters the concentration when filling a specific prescription. Suppose she alters the medication by adding water. a. Write a function
The pharmacist in Exercise 16 could also alter the medication by increasing the amount of pure medicine. a. Write a function that gives the concentration of the medication, c(x), as a function of the amount of pure medicine added in milliliters. b. What is the concentration if the pharmacist adds
Use graphs to determine which of these equations may be identities. a. cosx = xin(π/2 - x) b. cos x = sin (x - π/2) c. (csc x - cot x)(sec x + 1) = 1 d. tan x (cot x + tan x) = sec2x
Prove algebraically that the equation in Exercise 2d is an identity. In Exercise 2d tan x (cot x + tan x) = sec2x
In your own words, explain the difference between a trigonometric equation and a trigonometric identity.
A function f is even if f (-x) = f (x) for all x-values in its domain. It is odd if f (- x) = - f (x) for all x-values in its domain. Determine whether each function is even, odd, or neither. a. f (x) = sin x b. f (x) = cos x c. f (x) = tan x d. f (x) = cot x e. f (x) = sec x f. f (x) = csc x
In the next lesson, you'll see that cos 2A = cos2A - sin2A is an identity. Use this identity and the identities from this lesson to prove that a. cos 2A = 1 - 2 sin2 A b. cos 2A = 2 cos2 A - 1
Sketch the graph of y = 1/f(x) for each function.a.b.
Find another function that has the same graph as each function named below. More than one answer is possible a. y = co (π/2 -x) b. y = sin (π/2 -x) c. y = tan (π/2 -x) d. y = cos (- x) e. y = sin (- x) f. y = tan (- x) g. y = sin ( x + 2π) h. y = cos (π/2 -x) i. y = tan ( x + π)
Decide whether each expression is an identity by substituting values for A and B. a. cos (A + B) = cos A + cos B b. sin (A + B) = sin A + sin B c. cos (2A) = 2 cos A d. sin (2A) = 2 sin A
What is wrong with this statement?
Show that tan (A + B) is not equivalent to tan A + tan B. Then use the identities for sin (A + B) and cos (A + B) to develop an identity for tan (A + B).
You have seen that sin2A = 1 - cos2A and cos2A = 1 - sin2A. a. Use one of the double-angle identities to develop an expression that is equivalent to sin2A but does not contain the term cos2A. b. Use another double-angle identity to develop an expression equivalent to cos2A that does not contain
Set your graphing window to 0 ‰¤ x ‰¤ 4Ï€ and - 2 ‰¤ y ‰¤ 2.a. Graph equations in the form y = sin ax + sin bx, using the a- and b-values listed in the table. Record the period for each pair b. Explain how to find the period of any function
Set your graphing window to 0 ‰¤ x ‰¤ 24Ï€ and - 2 ‰¤ y ‰¤ 2.a. Graph equations in the form y = sinx/a + sin x/b, using the a- and b-values listed in the table. Record the period for each pair b. Explain how to find the period of any
When a tuning fork for middle C is struck, the resulting sound wave has a frequency of 262 cycles per second. The equation y = sin (262 · 2πx) is one possible model for this wave. a. Identify the period of this wave. Make a graph showing about five complete cycles. b. Suppose middle C on an
Find all values of that satisfy each equation. Use domain 0° ≤ θ < 360°. a. tan = 0.5317 b. sec = - 3.8637 c. csc = 1.1126 d. cot = - 4.3315
A fishing boat rides gently up and down, 10 times per minute, on the ocean waves. The boat rises and falls 1.5 m between each wave crest and trough. Assume the boat is on a crest at time 0 min. a. Sketch a graph of the boat's height above sea level over time. b. Use a cosine function to model your
Prove each identity. The sum and difference identities will be helpful. a. cos (2 - A) = cos A b. sin(3π/2 -A) = -cos A
Rewrite each expression with a single sine or cosine. a. cos 1.5 cos 0.4 + sin 1.5 sin 0.4 b. cos 2.6 cos 0.2 - sin 2.6 sin 0.2 c. sin 3.1 cos 1.4 - cos 3.1 sin 1.4 d. sin 0.2 cos 0.5 + cos 0.2 sin 0.5
Use identities to find the exact value of each expression. a. sin -11/π/12 b. sin 7π/12 c. tan π/12 d. cos π/8
Given π ≤ x ≤ 3π/2 and sin x = -2/3, find the exact value of sin 2x.
Use the identity for cos (A - B) and the identitiesAnd to prove that sin (A + B) = sin A cos B + cos A sin B
Use the identity for sin (A + B) from Exercise 6 to prove that sin (A - B) = sin A cos B - cos A sin B
Use the identity for sin (A + B) to prove the identity sin 2A = 2 sin A cos A.
Use the identity for cos (A + B) to prove the identity cos 2A = cos2A - sin2 A.
Find a connection between the graph of the polar equation r = a cos n and the graph of the associated rectangular equation y = a cos nx. Explain whether or not you can look at the graph of y = a cos nx and predict the shape and number of petals in the polar graph.
The graphs of polar equations in the forms r = a(cos θ + 1), r = a(cos θ - 1), r = a(sin θ + 1), and r = a(sin θ - 1) are called cardioids because they resemble hearts. Graph several curves in the cardioids family. Generalize your results by answering the questions. a. How do the graphs of r =
Write polar equations to create each graph. For 3b and d, you'll need more than one equation.a.b. c. d.
Think about what happens in a spiral and how the value of r changes as the value of Î¸ changes.a. Find an equation that creates a spiral. Check your work by graphing on your calculator with the domain 0° ‰¤ Î¸ ‰¤ 360°. What is the general form
In general, are polar equations functions? Explain your reasoning.

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