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mathematics
precalculus 1st
Precalculus 1st Edition Jay Abramson - Solutions
For the following exercises, rewrite the product as a sum or difference.16sin(16x)sin(11x)
For the following exercises, find all solutions exactly on the interval 0 ≤ θ < 2π.2sin θ = √3
Explain the effect of a damping factor on the graphs of harmonic motion functions.
For the half-angle formula given in the previous exercise for tan (x/2), explain why dividing by 0 is not a concern.
All of the Pythagorean identities are related. Describe how to manipulate the equations to get from sin2 t + cos2 t = 1 to the other forms.
Explain a situation where we would convert an equation from a product to a sum, and give an example.
For the following exercises, find all solutions exactly on the interval 0 ≤ θ < 2π.2sin θ = −√2
If we want to model cumulative rainfall over the course of a year, would a sinusoidal function be a good model? Why or why not?
Explain to someone who has forgotten the even-odd properties of sinusoidal functions how the addition and subtraction formulas can determine this characteristic for f(x) = sin(x) and g(x) = cos(x).
After examining the reciprocal identity for sec t, explain why the function is undefined at certain points.
Explain a situation where we would convert an equation from a sum to a product and give an example.
When solving linear trig equations in terms of only sine or cosine, how do we know whether there will be solutions?
What information is necessary to construct a trigonometric model of daily temperature? Give examples of two different sets of information that would enable modeling with an equation.
Is there only one way to evaluate cos (5π/4) ? Explain how to set up the solution in two different ways, and then compute to make sure they give the same answer.
Explain how to determine the double-angle formula for tan(2x) using the double-angle formulas for cos(2x) and sin(2x).
Examine the graph of f(x) = sec x on the interval [−π, π]. How can we tell whether the function is even or odd by only observing the graph of f(x) = sec x?
Explain two different methods of calculating cos(195°)cos(105°), one of which uses the product to sum. Which method is easier?
When solving a trigonometric equation involving more than one trig function, do we always want to try to rewrite the equation so it is expressed in terms of one trigonometric function? Why or why not?
Explain what types of physical phenomena are best modeled by sinusoidal functions. What are the characteristics necessary?
Explain the basis for the cofunction identities and when they apply.
Explain how to determine the reduction identities from the double-angle identity cos(2x) = cos2 x − sin2 x.
We know g(x) = cos x is an even function, and f(x) = sin x and h(x) = tan x are odd functions. What about G(x) = cos2 x, F(x) = sin2 x, and H(x) = tan2 x? Are they even, odd, or neither? Why?
Starting with the product to sum formula sin αcos β = 1/2 [sin(α + β) + sin(α − β)], explain how to determine the formula for cos αsin β.
Will there always be solutions to trigonometric function equations? If not, describe an equation that would not have a solution. Explain why or why not.
For the following exercises, find the function if sin t = x/x + 1.Suppose a 15-foot ladder leans against the side of a house so that the angle of elevation of the ladder is 42 degrees. How far is the foot of the ladder from the side of the house?
For the following exercises, find the function if sin t = x/x + 1.A 20-foot ladder leans up against the side of a building so that the foot of the ladder is 10 feet from the base of the building. If specifications call for the ladder’s angle of elevation to be between 35 and 45 degrees, does the
For the following exercises, find the function if sin t = x/x + 1.What percentage grade should a road have if the angle of elevation of the road is 4 degrees? (The percentage grade is defined as the change in the altitude of the road over a 100-foot horizontal distance. For example a 5% grade means
For the following exercises, find the function if sin t = x/x + 1.The line y = −3/7 x passes through the origin in the x,y-plane. What is the measure of the angle that the line makes with the negative x-axis?
For the following exercises, find the function if sin t = x/x + 1.The line y = 3/5 x passes through the origin in the x,yplane. What is the measure of the angle that the line makes with the positive x-axis?
For the following exercises, find the function if sin t = x/x + 1.A truss for the roof of a house is constructed from two identical right triangles. Each has a base of 12 feet and height of 4 feet. Find the measure of the acute angle adjacent to the 4-foot side.
For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input csc x as 1/sin x.A video camera is focused on a rocket on a launching pad 2 miles from the camera. The
For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input csc x as 1/sin x.Standing on the shore of a lake, a fisherman sights a boat far in the distance to his
For the following exercises, find the function if sin t = x/x + 1.Without using a calculator, approximate the value of arctan(10,000). Explain why your answer is reasonable.
For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input csc x as 1/sin x.A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in
For the following exercises, find the function if sin t = x/x + 1.An isosceles triangle has two congruent sides of length 9 inches. The remaining side has a length of 8 inches. Find the angle that a side of 9 inches makes with the 8-inch side.
For the following exercises, find the exact value without the aid of a calculator. sin (cos 08-¹( ¹(x + 1))
For the following exercises, determine whether the equation is true or false.he grade of a road is 7%.This means that for every horizontal distance of 100 feet on the road, the vertical rise is 7 feet. Find the angle the road makes with the horizontal in radians.
For the following exercises, find the exact value without the aid of a calculator.Graph f(x) = cos x and f(x) = sec x on the interval [0, 2π) and explain any observations.
For the following exercises, find the exact value of the expression in terms of x with the help of a reference triangle.tan (sin−1 (x +1/2))
For the following exercises, let f(x) = cos x.On [0, 2π), solve f(x) = 1/2.
For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why. -1 -¹ (sin(57)) sin
For the following exercises, find the period and horizontal shift of each function.g(x) = 3tan(6x + 42)
For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why. tan -¹ (sin(-57))
For the following exercises, use a calculator to evaluate each expression. Express answers to the nearest hundredth.tan−1(6)
For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.f(x) = −3cot(2x)
For the following exercises, find the exact value without the aid of a calculator.tan−1 (−1)
For the following exercises, let f(x) = 3/5 cos(6x).Where is the function increasing on the interval [0, 2π]?
For the following exercises, find the exact value without the aid of a calculator. cos 1 (2)
For the following exercises, let f(x) = sin x. On [0, 2π), solve f(x) = 0.
For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.f(x) = 7sec(5x)
For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why. os(sin-¹())
For the following exercises, find and graph one period of the periodic function with the given amplitude, period, and phase shift.Sine curve with amplitude 3, period π/3 , and phase shift (h, k) = (π/4 ,2)
For the following exercises, find the exact value without the aid of a calculator. √3 -(-V³) 2 sin-1
For the following exercises, let f(x) = sin x.On [0, 2π), solve f(x) = 1/2 .
For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes.f(x) = 9/10 csc(πx)
For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why. 3 sin (cos-¹())
For the following exercises, find the exact value without the aid of a calculator. s-(tan (³7)) 4 COS
For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes. f(x) = 2csc (x + 7) -1 4
For the following exercises, find and graph one period of the periodic function with the given amplitude, period, and phase shift.Cosine curve with amplitude 2, period π/6, and phase shift (h,k) = (−π/4 ,3)
For the following exercises, find the exact value without the aid of a calculator. (()), 9 sin
For the following exercises, let f(x) = sin x.Evaluate f (π/2).
For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why. sin tan 4 -¹({})) 3
For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes. f(x) = -sec(x. π 3 - 2
For the following exercises, graph the function. Describe the graph and, wherever applicable, any periodic behavior, amplitude, asymptotes, or undefined points.f(x) = 5cos(3x) + 4sin(2x)
For the following exercises, find the exact value without the aid of a calculator. 3 sin (sec-¹(²)) 5
For the following exercises, let f(x) = sin x.On [0, 2π), f(x) = √2/2 . Find all values of x.
For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why. 12 cos (tan-¹(13))
For the following exercises, find the exact value without the aid of a calculator. 3 cot (sin-¹())
For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes. 7 f(x) = 1/-csc(x - 1) 5 4
For the following exercises, graph the function. Describe the graph and, wherever applicable, any periodic behavior, amplitude, asymptotes, or undefined points.f(x) = esint
For the following exercises, let f(x) = sin x.On [0, 2π), the maximum value(s) of the function occur(s) at what x-value(s)?
For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes. f(x) = 5(cot(x + 1)-3) 2
For the following exercises, find the exact value without the aid of a calculator. tan cos 5 (1/3))
For the following exercises, find the exact value, if possible, without a calculator. If it is not possible, explain why. os (sin-¹(1)) 2 cos
For the following exercises, let f(x) = sin x.On [0, 2π), the minimum value(s) of the function occur(s) at what x-value(s)?
For the following exercises, find the exact value. tan−1 (√3)
For the following exercises, let f(x) = sin x.Show that f(−x) = −f(x). This means that f(x) = sin x is an odd function and possesses symmetry with respect to ________________.
For the following exercises, find the exact value of the expression in terms of x with the help of a reference triangle.tan(sin−1 (x − 1))
For the following exercises, find and graph two periods of the periodic function with the given stretching factor, ∣A∣, period, and phase shift.A tangent curve, A = 1, period of π/3 ; and phase shift (h,k) = (π/4 , 2)
For the following exercises, let f(x) = cos x.On [0, 2π), solve the equation f(x) = cos x = 0.
For the following exercises, find the exact value of the expression in terms of x with the help of a reference triangle.sin(cos−1 (1 − x))
For the following exercises, find and graph two periods of the periodic function with the given stretching factor, ∣A∣, period, and phase shift.A tangent curve, A = −2, period of π/4 , and phase shift (h,k) = (−π/4 ,−2)
For the following exercises, find the exact value of the expression in terms of x with the help of a reference triangle. cos (sin-¹())
For the following exercises, find an equation for the graph of each function. X = - x= f(x) 10- -6 |-10- 얘 2 Y 프2 X = x
For the following exercises, find the exact value without the aid of a calculator.Graph f(x) = sin x and f(x) = csc x and explain any observations.
For the following exercises, let f(x) = cos x.On [0, 2π), find the x-intercepts of f(x) = cos x.
For the following exercises, find the exact value of the expression in terms of x with the help of a reference triangle.cos(tan−1 (3x − 1))
For the following exercises, find an equation for the graph of each function. I x= -1 -0.5 f(x) 8 4- -8 0.5 x= 1
For the following exercises, find the exact value without the aid of a calculator.Graph the function f(x) = x 1 − x3/3! + x5/5! − x7/7! on the interval [−1, 1] and compare the graph to the graph of f(x) = sin x on the same interval. Describe any observations.
For the following exercises, let f(x) = cos x.On [0, 2π), find the x-values at which the function has a maximum or minimum value.
For the following exercises, find an equation for the graph of each function. X= f(x) 10- 2 1 LAN -6+ -10 x = 7 KIN x EIN X
For the following exercises, find the exact value. cos−1 (−0.4)
For the following exercises, let f(x) = cos x.On [0, 2π), solve the equation f(x) = √3/2.
For the following exercise, evaluate the expression without using a calculator. Give the exact value. -(1) - V3 2 sin COS COS -(1/²). 2 sin V2 2 - + sin اسکان cos ¹(1) sin ¹(0) + cos +-(-/-) (1²) - 2
For the following exercises, find an equation for the graph of each function. X= -품 5π f(x) 10- TT 8 lo -10 x Зл 8 5m 8 X
For the following exercises, find the exact value.cos(tan−1 (x2))
For the following exercises, let f(x) = cos x.Graph h(x) = x + sin x on [0, 2π]. Explain why the graph appears as it does.
For the following exercises, find an equation for the graph of each function. x=-27 I | I I 1 f(x) 10 2 -6 -10- x=-π x = π I I 1 1 I x = 2π X
For the following exercises, find the function if sin t = x/x + 1.cos t
For the following exercises, suppose sin t = x/x + 1,tan t
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