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mathematics
precalculus

Questions and Answers of Precalculus

Find the exact value of each of the remaining trigonometric functions of θ. cos 0 = 1 tan 0 > 0
Graph the function. Graph should contain at least two periods. Use the graph to determine the domain and the range of function. y = -tan x 2 - |
Find the exact value of each of the remaining trigonometric functions of θ. sin 0 = 2 3' tan 0 < 0
Graph each function. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function. y = 4 sin %3D
Graph each function. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function. y = 5 cos(πx) -3
To approximate the speed of the current of a river, a circular paddle wheel with radius 4 feet is lowered into the water. If the current causes the wheel to rotate at a speed of 10 revolutions per
A spin balancer rotates the wheel of a car at 480 revolutions per minute. If the diameter of the wheel is 26 inches, what road speed is being tested? Express your answer in miles per hour. At how
At the Cable Car Museum you can see the four cable lines that are used to pull cable cars up and down the hills of San Francisco. Each cable travels at a speed of 9.55 miles per hour, caused by a
Naples, Florida, is approximately 90 miles due west of Ft. Lauderdale. How much sooner would a person in Ft. Lauderdale first see the rising Sun than a person in Naples?Consult the figure. When a
How fast would you have to travel on the surface of Earth at the equator to keep up with the Sun (that is, so that the Sun would appear to remain in the same position in the sky)?
A nautical mile equals the length of arc subtended by a central angle of 1 minute on a great circle on the surface of Earth. See the figure. If the radius of Earth is taken as 3960 miles, express 1
Eratosthenes of Cyrene (276–195 BC) was a Greek scholar who lived and worked in Cyrene and Alexandria. One day while visiting in Syene he noticed that the Sun’s rays shone directly down a well.
For a 60-foot Little League Baseball field, the distance from home base to the nearest fence (or other obstruction) in fair territory should be a minimum of 200 feet. The commissioner of parks and
Two pulleys, one with radius r1 and the other with radius r2, are connected by a belt. The pulley with radius r1 rotates at ω1 revolutions per minute, whereas the pulley with radius r2 rotates at
Which angle has the larger measure: 1 degree or 1 radian? Or are they equal?
Explain the difference between linear speed and angular speed.
For a circle of radius r, a central angle of θ degrees subtends an arc whose length s is s = π/180 rθ. Discuss whether this is a true or false statement. Give reasons to defend your position.
Establish the identity: (sin θ cos ϕ)2 + (sin θ sin ϕ)2 + cos2 θ = 1
Prove the quotient identities given in formula (3).
Prove the reciprocal identities given in formula (2).
Show that the period of f(θ) = tan θ is π.
Show that the period of f(θ) = csc θ is 2π.
Show that the period of f(θ) = sec θ is 2π.
Show that the period of f(θ) = cos θ is 2π.
Show that the period of f(θ) = sin θ is 2π.Assume that 0 < p < 2π exists so that sin(θ + p) sin θ for all θ. Let θ = 0 to find p. Then let θ = π/2 to obtain a contradiction.
Show that the range of the cotangent function is the set of all real numbers.
Show that the range of the tangent function is the set of all real numbers.
Two oceanfront homes are located 8 miles apart on a straight stretch of beach, each a distance of 1 mile from a paved path that parallels the ocean. Sally can jog 8 miles per hour on the paved path,
From a parking lot, you want to walk to a house on the beach. The house is located 1500 feet down a paved path that parallels the ocean, which is 500 feet away. See the illustration. Along the path
If f(θ) = csc θ and f(a) = 2, find the exact value of:(a) f(-a)(b) f(a) + f(a + 2π) + f(a + 4π)Use the periodic and even–odd properties.
If f(θ) = sec θ and f(a) = -4, find the exact value of:(a) f(-a)(b) f(a) + f(a + 2π) + f(a + 4π)Use the periodic and even–odd properties.
If f(θ) = cot θ and f(a) = -3, find the exact value of:(a) f (-a)(b) f(a) + f (a + π) + f (a + 4π)Use the periodic and even–odd properties.
If f(θ) = tan θ and f(a) = 2, find the exact value of:(a) f(-a)(b) f(a) + f(a + π) + f(a + 2π)Use the periodic and even–odd properties.
If f(θ) = cos θ and f(a) = 1/4, find the exact value of:(a) f(-a)(b) f(a) + f(a + 2π) + f(a – 2π)Use the periodic and even–odd properties.
If f(θ) = sin θ and f(a) = 1/3 find the exact value of:(a) f(-a)(b) f(a) + f(a + 2π) + f(a + 4π)Use the periodic and even–odd properties.
Is the cosecant function even, odd, or neither? Is its graph symmetric? With respect to what?
Is the secant function even, odd, or neither? Is its graph symmetric? With respect to what?
Is the cotangent function even, odd, or neither? Is its graph symmetric? With respect to what?
Is the tangent function even, odd, or neither? Is its graph symmetric? With respect to what?
In problem, use the figure to approximate the value of the six trigonometric functions at t to the nearest tenth. Then use a calculator to approximate each of the six trigonometric functions at t.(a)
In problem, use the figure to approximate the value of the six trigonometric functions at t to the nearest tenth. Then use a calculator to approximate each of the six trigonometric functions at t.(a)
If θ, 0 < θ < π, is the angle between the positive x-axis and a nonhorizontal, nonvertical line L, show that the slope m of L equals tan θ. The angle θ is called the inclination of L. (cos
An object is propelled upward at an angle θ, 45° < θ < 90°, to the horizontal with an initial velocity of υ0 feet per second from the base of an inclined plane that makes an angle of 45°
A designer of decorative art plans to market solid gold spheres encased in clear crystal cones. Each sphere is of fixed radius R and will be enclosed in a cone of height h and radius r. See the
Two oceanfront homes are located 8 miles apart on a straight stretch of beach, each a distance of 1 mile from a paved road that parallels the ocean. See the figure.Sally can jog 8 miles per hour
In a certain piston engine, the distance x (in centimeters) from the center of the drive shaft to the head of the piston is given by the functionwhere θ is the angle between the crank and the path
See the figure.If friction is ignored, the time t (in seconds) required for a block to slide down an inclined plane is given by the functionwhere a is the length (in feet) of the base and g ≈ 32
The projectile is fired at an angle of 50° to the horizontal with an initial speed of 200 feet per second.In problem, find the range R and maximum height H.The path of a projectile fired at an
The projectile is fired at an angle of 25° to the horizontal with an initial speed of 500 meters per second.In problem, find the range R and maximum height H.The path of a projectile fired at an
The projectile is fired at an angle of 30° to the horizontal with an initial speed of 150 meters per second.In problem, find the range R and maximum height H.The path of a projectile fired at an
The projectile is fired at an angle of 45° to the horizontal with an initial speed of 100 feet per second.In problem, find the range R and maximum height H.The path of a projectile fired at an
Use a calculator in radian mode to complete the following table.What can you conclude about the value of g(θ) = cos θ – 1/ θ as θ approaches 0? 0.0001 0.00001 0.01 0.001 0.5 0.4 0.2 0.1 1 cos e
Use a calculator in radian mode to complete the following table.What can you conclude about the value of f(θ) = sin θ/ θ as θ approaches 0? 0.0001 0.5 0.2 0.1 0.01 0.001 0.00001 0.4 sin e sin 0 f
Find two negative and three positive angles, expressed in radians, for which the point on the unit circle that corresponds to each angle is (-√2/2, √2/2).
Find two negative and three positive angles, expressed in radians, for which the point on the unit circle that corresponds to each angle is(1/2, √3/2).
(a) Find g(π/6). What point is on the graph of g?(b) Assuming g is one-to-one*, use the result of part (a) to find a point on the graph of g-1.(c) What point is on the graph of = 28(x-) if x=
(a) Find f(π/4). What point is on the graph of f?(b) Assuming f is one-to-one*, use the result of part (a) to find a point on the graph of f-1.(c) What point is on the graph of y = f(x + π/4) - 3
In problem, f(x) = sin x, g(x) = cos x, h(x) = 2x, and p(x) = x/2. Find the value of each of the following:(h ° f)(5π/6)
In problem, f(x) = sin x, g(x) = cos x, h(x) = 2x, and p(x) = x/2. Find the value of each of the following:(p ° g)(315°)
In problem, f(x) = sin x, g(x) = cos x, h(x) = 2x, and p(x) = x/2. Find the value of each of the following:(g ° p)(60°)
In problem, f(θ) = sin θ and g(θ) = cos θ. Find the exact value of each function below if θ = 60°. Do not use a calculator.g(θ)
In problem, f(θ) = sin θ and g(θ) = cos θ. Find the exact value of each function below if θ = 60°. Do not use a calculator.f(θ)
If cos θ = 2/3, find sec θ.
If sin θ = 1/5, find csc θ.
If f(θ) = cot θ = -2, find f(θ + π).
If f(θ) = tan θ = 3, find f(θ + π).
If f(θ) = cos θ = 0.3, find f (θ + π).
If f(θ) = sin θ = 0.1, find f (θ + π).
Find the exact value of:tan 40° + tan 140°
Find the exact value of:sin 40° + sin 130° + sin 220° + sin 310°
Find the exact value of:tan 60° + tan 150°
Find the exact value of:sin 45° + sin 135° + sin 225° + sin 315°
A point on the terminal side of an angle θ in standard position is given. Find the exact value of each of the six trigonometric functions of θ.(0.3, 0.4)
A point on the terminal side of an angle θ in standard position is given. Find the exact value of each of the six trigonometric functions of θ. (1/3, 1/4)
A point on the terminal side of an angle θ in standard position is given. Find the exact value of each of the six trigonometric functions of θ.(-1, 1)
A point on the terminal side of an angle θ in standard position is given. Find the exact value of each of the six trigonometric functions of θ.(-2, -2)
A point on the terminal side of an angle θ in standard position is given. Find the exact value of each of the six trigonometric functions of θ.(-1, -2)
A point on the terminal side of an angle θ in standard position is given. Find the exact value of each of the six trigonometric functions of θ.(2, -3)
A point on the terminal side of an angle θ in standard position is given. Find the exact value of each of the six trigonometric functions of θ.(5, -12)
A point on the terminal side of an angle θ in standard position is given. Find the exact value of each of the six trigonometric functions of θ.(-3, 4)
Use a calculator to find the approximate value of each expression rounded to two decimal places.tan 1°
Use a calculator to find the approximate value of each expression rounded to two decimal places.sin 1°
Use a calculator to find the approximate value of each expression rounded to two decimal places.tan 1
Use a calculator to find the approximate value of each expression rounded to two decimal places.sin 1
Use a calculator to find the approximate value of each expression rounded to two decimal places.csc 5π/13
Use a calculator to find the approximate value of each expression rounded to two decimal places.cot π/12
Use a calculator to find the approximate value of each expression rounded to two decimal places.sin π/8
Find the amplitude, period, and phase shift of function. Graph function. Show at least two periods.y = 2 sin(2x – π)
Find the amplitude, period, and phase shift of function. Graph function. Show at least two periods. y = 2 cos х
Find the amplitude, period, and phase shift of function. Graph function. Show at least two periods.y = 4 sin(3x)
Determine the amplitude and period of function without graphing.y = -2 cos(3πx)
Determine the amplitude and period of function without graphing. y = -8 sin X. 2
Determine the amplitude and period of function without graphing.y = sin(2x)
Determine the amplitude and period of function without graphing.y = 4 cos x
Graph the function. Graph should contain at least two periods. Use the graph to determine the domain and the range of function. TT y = 5 cot %3D 4
Graph the function. Graph should contain at least two periods. Use the graph to determine the domain and the range of function. y = 4 tan
Graph the function. Graph should contain at least two periods. Use the graph to determine the domain and the range of function.y = 3 cos(4x + 2) + 1
Graph the function. Graph should contain at least two periods. Use the graph to determine the domain and the range of function.y = 4 sin(2x + 4) - 2
Graph the function. Graph should contain at least two periods. Use the graph to determine the domain and the range of function. y = csc x + 4 4.
Graph the function. Graph should contain at least two periods. Use the graph to determine the domain and the range of function.y = 4 sec (2x)

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