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mathematics
calculus

Questions and Answers of Calculus

The tide, or depth of the ocean near the shore, changes throughout the day. The water depth d (in feet) of a bay can be modeled by d = 35 - 28 cos π / 6.2 t where t represents the time in hours,
The heights h (in inches) of pistons 1 and 2 in an automobile engine can be modeled by h1 = 3.75 sin 733t + 7.5 and h2 = 3.75 sin 733 (t + 4 π / 3) + 7.5 respectively, where t is measured in
The index of refraction n of a transparent material is the ratio of the speed of light in a vacuum to the speed of light in the material. Some common materials and their indices of refraction are air
(a) Write a sum formula for sin (u + v + w). (b) Write a sum formula for tan (u + v + w).
Find the solution of each inequality in the interval [0, 2π). a. sin x ≥ 0.5 b. cos x ≤ − 0.5 c. tan x < sin x d. cos x ≥ sin x
Consider the function f (x) = sin4 x + cos4 x (a) Use the power-reducing formulas to write the function in terms of cosine to the first power. (b) Determine another way of rewriting the original
Verify that for all integers n, sin [(12n + 1) π / 6] = 1 / 2.
A sound wave is modeled byWhere pn(t) = 1 / n sin (524n Ï€ t), t represent the time (in seconds). (a) Find the sine components pn(t) and use a graphing utility to graph the components. Then
Three squares of side length s are placed side by side (see figure). Make a conjecture about the relationship between the sum u + v and w. Prove your conjecture by using the identity for the tangent
The path traveled by an object (neglecting air resistance) that is projected at an initial height of h0 feet, an initial velocity of v0 feet per second, and an initial angle is given bywhere the
The length of each of the two equal sides of an isosceles triangle is 10 meters (see figure). The angle between the two sides is Î¸.(a) Write the area of the triangle as a function of
Use the figure to derive the formulas for sin θ / 2, cos θ / 2, and tan θ / 2 where θ is an acute angle.
The force F (in pounds) on a person's back when he or she bends over at an angle from an upright position is modeled bywhere W represents the person's weight (in pounds).(a) Simplify the model.(b)
1. An equation that is true for all real values in the domain of the variable is an ________. 2. An equation that is true for only some values in the domain of the variable is a ________ ________.
In Exercises 1-4, verify the identity algebraically. Use the table feature of a graphing utility to check your result numerically. 1. 1 / tan x + 1 / cot x = tan x + cot x 2 .1 / sin x - 1 / csc x =
In Exercises 1-4, verify the identity algebraically. Use a graphing utility to check your result graphically. 1. sec y cos y = 1 2. cot2 y(sec2 y − 1) = 1 3. tan2 Θ / sec Θ = sin Θ tan Θ 4.
In Exercises 1-8, fill in the blank to complete the fundamental trigonometric identity. 1. 1 / cot u = __________ 2. cos u / sin u =__________ 3. cos (π / 2 - u) = __________ 4. 1 + ________ = csc2
In Exercises 1-4, verify the identity by converting the left side into sines and cosines. 1. cot2 t / csc t = 1 - sin2 t / sin t 2. cos x + sin x tan x = sec x 3. secx cos x = sin x tan x 4. cot x
In Exercises 1-4, verify the identity. 1. sin 1/2 x cos x - sin 5/2 x cos x = cos3 x √sin x 2. sec6 x(sec x tan x) - sec4 x(sec x tan x) = sec5 x tan3 3. (1 + sin y) [1 + sin(-y)] = cos2y 4. tan x
In Exercises 1 and 2, describe the error(s). 1. 1 / tan x + cot(-x) = cot x + cot x = 2 cot x 2. 1 + sec(-Θ) / sin(- Θ) + tan(-Θ) = 1 - sec Θ / sin Θ - tan Θ
In Exercises 1-4, (a)use a graphing utility to grapheach side of the equation to determine whether the equation is an identity, (b)use the table feature of the graphing utility to determine
In Exercises 1-4, verify the identity. 1. tan5 x = tan3 x sec2 x − tan3 x 2. sec4 x tan2 x = (tan2 x + tan4 x)sec2 x 3. cos3 x sin2 x = (sin2 x − sin4 x)cos x 4. sin4 x + cos4 x = 1 − 2 cos2 x
In Exercises 1 and 2, use the cofunction identities to evaluate the expression without using a calculator. 1. sin2 25° + sin2 65° 2. tan2 63° + cot2 16° − sec2 74° − csc2 27°
In Exercises 1-2, verify the identity. 1. tan(sin -1x) = x / √1 - x2 2. cos(sin -1x) = √1 - x2 3. tan(sin -1 x - 1 / 4) = x - 1 / √16 - (x - 1)2 4. tan(cos -1x + 1 / 2) = √4 - (x + 1)2 / x +
The rate of change of the function f (x) = sin x + csc x is given by the expression cos x − csc x cot x. Show that the expression for the rate of change can also be written as −cos x cot2 x.
The length s of a shadow cast by a vertical gnomon (a device used to tell time) of height h when the angle of the sun above the horizon is can be modeled by the equation s = 4 sin(90o Î˜)
In Exercises 1-3, determine whether the statement is true or false. Justify your answer. 1. tan x2 = tan2 x 2. cos(Θ - π / 2) = sin Θ 3. The equation sin2 + cos2 = 1 + tan2 is an identity
In Exercises 1-4, explain why the equation is not an identity and find one value of the variable for which the equation is not true. 1. sin Θ = √1 - cos2 Θ 2. tan Θ = √sec2 Θ -1 3. 1 - cos
In Exercises 1-4, verify the identity. 1. tan t cot = 1 2. tan x cot x / cos x =sec x 3. (1 + sin α)(1 − sin α) = cos2 α 4. cos2β − sin2β = 2 cos2 β - 1 5. cos2β − sin2 β = 1 − 2
1. When solving a trigonometric equation, the preliminary goal is to ________ the trigonometric function on one side of the equation. 2. The ________ solution of the equation 2 sin θ +1 = 0 is θ =
1. Explain what happens when you divide each side of the equation cot x cos2 x = 2 cot x by cot x. Is this a correct method to use when solving equations?2. Explain how to use the figure to solve the
Use a graphing utility to confirm the solutions found in Example 6 in two different ways.(a) Graph both sides of the equation and find the x-coordinates of the points at which the graphs
Solve the equation. 1. √3 csc x - 2 = 0 2. tan x + √3 = 0 3. cos x + 1 = -cos x 4. 3 sin x + 1 = sin x 5. 3 sec2 x - 4 = 0
Find all solutions of the equation in the interval (0, 2π). 1. sin x - 2 = cos x - 2 2. cos x + sin x tan x = 2 3. 2 sin2x = 2 + cos x 4. tan2x = sec x -1 5. sin2x = 3 cos2x 6. 2 sec2x + tan2 x - 3
Solve the multiple-angle equation. 1. 2 cos 2x − 1 = 0 2. 2 sin 2x + √3 = 0 3. tan 3x - 1 = 0 4. sec 4x - 2 = 0 5. 2 cos x / 2 - √2 = 0
Find the x-intercepts of the graph.1. y = sin Ï€x / 2 + 12. y = sin Ï€x + cos Ï€x
Use a graphing utility to approximate (to three decimal places) the solutions of the equation in the interval (0, 2π). 1. 5 sin x + 2 = 0 2. 2tan x + 7 = 0 3. sin x − 3 cos x = 0 4. sin x + 4
Verify that each x-value is a solution of the equation. 1. tan x - √3 = 0 (a) x = π / 3 (b) x = 4π / 3 2. sec x -2 = 0 (a) x = π / 3 (b) x = 5π / 3 3. 3 tan2 2x - 1 = 0 (a) x = π / 12 (b) x =
Solve the equation. 1. tan2 x + tan x − 12 = 0 2. tan2 x − tan x − 2 = 0 3. sec2 x − 6 tan x = −4 4. sec2 x + tan x = 3 5. 2 sin2 x + 5 cos x = 4
Use the Quadratic Formula to find all solutions of the equation in the interval (0, 2π). Round your result to four decimal places. 1. 12 sin2 x − 13 sin x + 3 = 0 2. 3 tan2 x + 4 tan x − 4 = 0
Use a graphing utility to approximate (to three decimal places) the solutions of the equation in the given interval. 1. 3 tan2 x + 5 tan x − 4 = 0, [- π / 2, π / 2] 2. cos2 x − 2 cos x − 1 =
(a) Use a graphing utility to graph the function and approximate the maximum and minimum points on the graph in the interval [0, 2π), and (b) solve the trigonometric equation and verify that its
Use the graph to approximate the number of points of intersection of the graphs of y1 and y2.1. y1 = 2 sin xy2 = 3x + 12. y1 = 2 sin x y2 = 1 / 2 x + 1
Consider the function f(x) sin x / x and its graph, shown in the figure below.(a) What is the domain of the function? (b) Identify any symmetry and any asymptotes of the graph. (c) Describe the
Consider the function f(x) = cos 1 / x and its graph, shown in the figure below.(a) What is the domain of the function? (b) Identify any symmetry and any asymptotes of the graph. (c) Describe the
A weight is oscillating on the end of a spring (see figure). The displacement from equilibrium of the weight relative to the point of equilibrium is given by y = 1/12 (cos 8t - 3 sin 8t) where y is
1. The displacement from equilibrium of a weight oscillating on the end of a spring is given by y = 1.56e-0.22t cos 4.9t Where y is the displacement (in feet) and t is the time (in seconds). Use a
A baseball is hit at an angle of Î¸ with the horizontal and with an initial velocity of v0 = 100 feet per second. An outfielder catches the ball 300 feet from home plate (see figure).
The table shows the normal daily high temperatures C in Chicago (in degrees Fahrenheit) for month t, with t = 1 corresponding to January. Months, t .............................................
The height h (in feet) above ground of a seat on a Ferris wheel at time t (in minutes) can be modeled by h (t) = 53 + 50 sin (π / 16t - π/2). The wheel makes one revolution every 32 seconds. The
The area of a rectangle inscribed in one arc of the graph of y = cos x (see figure) is given by A = 2x cos x, 0(a) Use a graphing utility to graph the area function, and approximate the area of the
Consider the function f (x) = 3 sin (0.6x − 2). (a) Approximate the zero of the function in the interval [0, 6]. (b) A quadratic approximation agreeing with f at x = 5 is g (x) = -0.45x2 + 5.52x -
Find the least positive fixed point of the function f. [A fixed point of a function f is a real number c such that f (c) = c.] 1. f (x) = tan (πx / 4) 2. f(x) = cos x
1. The equation 2 sin 4t − 1 = 0 has four times the number of solutions in the interval [0, 2π) as the equation 2 sin t − 1 = 0. 2. The trigonometric equation sin x = 3.4 can be solved using an
1. sin(u − v) = ______ 2. cos(u + v) = ______ 3. tan(u + v) = ______ 4. sin(u + v) = ______ 5. cos(u − v) = ______ 6. tan(u − v) = ______
Write a proof of the formula for sin(u + v). Write a proof of the formula for sin(u − v).
Let x = Ï€ / 3 in the identity in Example 8 and define the functions f and g as follows.(a) What are the domains of the functions f and g? (b) Use a graphing utility to complete the
In Exercises 1-4, find the exact values of the sine, cosine, and tangent of the angle. 1. 11 π / 12 = 3 π / 4 + π / 6 2. 7 π / 12 = π / 3 + π / 4 3. 17 π / 12 = 9 π / 4 - 5 π / 6 4. - π
In Exercises 1-4, write the expression as the sine, cosine, or tangent of an angle. 1. sin 3 cos 1.2 − cos 3 sin 1.2 2. cos π / 7 cos π / 5 - sin π / 7 sin π / 5 3. sin 60° cos 15° + cos
In Exercises 1-4, find the exact value of the expression. 1. sin π / 12 cos π / 4 + cos π / 12 sin π / 4 2. cos π / 16 cos 3 π / 16 - sin π / 16 sin 3 π / 16 3. cos 130° cos 10° + sin
In Exercises 1-4, find the exact value of the trigonometric expression given that sin u = - 3 / 5, where 3 π / 2 < u < 2 π, and cos v = 15 / 17, where 0 < v < π / 2. 1. sin(u + v) 2. cos(u −
In Exercises 1-4, find the exact value of the trigonometric expression given that sin u = - 7 / 25 and cos v = - 4 / 5.(both u and v are in Quadrant III). 1. cos(u + v) 2. sin(u + v) 3. tan(u −
In Exercises 1-4, write the trigonometric expression as an algebraic expression. 1. sin(arcsin x + arccos x) 2. sin(arctan 2x − arccos x) 3. cos(arccos x + arcsin x) 4. cos(arccos x − arctan x)
In Exercises 1-4, verify the identity. 1. sin(π / 2 - x) = cos x 2. sin(π / 2 - x) = cos x 3. sin(π / 6 + x) = 1 / 2(cos + √3 sin x) 4. cos(5 π / 4 - x) = -√2 / 2(cos x + sin x)
In Exercises 1 - 4, write the expression as a trigonometric function of only Θ, and use a graphing utility to confirm your answer graphically. 1. cos(3π / 2 - Θ) 2. sin(π + Θ) 3. csc(3π / 2 +
In Exercises 1-4, find all solutions of the equation in the interval [0, 2π). 1. sin(x + ) − sin x + 1 = 0 2. cos(x + ) − cos x − 1 = 0 3. cos(x + π / 4) - cos(x - π / 4) = 1 4. sin( x +
In Exercises 1-4, find the exact value of each expression. 1. a. cos (π / 4 + π / 3) b. cos π / 4 + cos π / 3 2. a. sin(7π / 6 - π / 3) b. sin 7π / 6 - sin π / 3 3. a. sin(135° − 30°)
In Exercises 1-4, use a graphing utility to approximate the solutions of the equation in the interval [0, 2π). 1. cos(x + π / 4) + cos(x - π / 4) = 1 2. tan(x + π) - cos(x + π / 2) = 0 3. sin(x
A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is
The equation of a standing wave is obtained by adding the displacements of two waves traveling in opposite directions (see figure). Assume that each of the waves has amplitude A, period T, and
In Exercises 1-4, determine whether the statement is true or false. Justify your answer. 1. sin(u ± v) = sin u cos v ± cos u sin v 2. cos(u ± v) = cos u cos v ± sin u sin v 3. When α and β are
Describe the error. tan (x - π / 4) = tan x - tan(x / 4) / 1 - tan x tan(π / 4) = tan x - 1 / 1 - tan x = - 1
In Exercises 1-4, verify the identity. 1. cos(nπ + Θ) = (−1)n cos Θ, n is an integer 2. sin(nπ + Θ) = (−1)n sin Θ, n is an integer 3. a sin BΘ + b cos BΘ = √a2 + b2 sin(BΘ + C),
In Exercises 1-2, use the formulas given in following Exercises to write the trigonometric expression in the following forms. a. √a2 + b2 sin(BΘ + C) b. √a2 + b2 cos(BΘ - C) 1. sin Θ + cos Θ
In Exercises 1 and 2, use the formulas given in following Exercises to write the trigonometric expression in the form a sin BΘ + b cos BΘ. 1. 2 sin[Θ + (π / 4)] 2. 5 cos[Θ - (π / 4)]
In Exercises 1 and 2, use the figure, which shows two lines whose equations are y1 = m1x + b1 and y2 = m2x + b2. Assume that both lines have positive slopes. Derive a formula for the angle between
In Exercises 1 and 2, use a graphing utility to graph y1 and y2 in the same viewing window. Use the graphs to determine whether y1 = y2. Explain your reasoning. 1. y1 = cos(x + 2), y2 = cos x + cos
Fill in the blanks 1. sin 2u = ______ 2. cos 2u = ______ 3. sin u cos v = ______ 4. 1 --- cos 2u / 1 + cos 2u = __________ 5. sin u / 2 =____________ 6. cos u − cos v = ______
In Exercises 1-4, use a double-angle formula to rewrite the expression. 1. 6 sin x cos x 2. sin x cos x 3. 6 cos2 x − 3 4. cos2 x 1 / 2
In Exercises 1-2, use the given conditions to find the exact values of sin 2u, cos 2u, and tan 2u using the double-angle formulas. 1. sin u = −3 / 5, 3π / 2 < u < 2π 2. cos u = −4 / 5, π /
Rewrite cos 4x in terms of cos x.
Rewrite tan 3x in terms of tan x.
In Exercises 1-2, use the power-reducing formulas to rewrite the expression in terms of first powers of the cosines of multiple angles. 1. cos4 x 2. sin8 x
In Exercises 1-4, use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. 1. 75° 2. 165° 3. 112° 30o 4. 67° 330o
In Exercises 1-4, use the given conditions to (a) determine the quadrant in which u / 2 lies, and (b) find the exact value of sin(u / 2), cos(u / 2), and tan(u / 2) using the half-angle
In Exercises 1-2, find all solutions of the equation in the interval [0, 2π). Use a graphing utility to graph the equation and verify the solutions. 1. sin x / 2 + cos x = 0 2. sin x / 2 + cos x -
Formulas In Exercises 1-2, use the product-to-sum formulas to rewrite the product as a sum or difference 1. sin 5Θ sin 3 Θ 2. 7 cos( 5β ) sin 3β 3. cos 2Θ cos 4Θ 4. sin(x + y) cos(x − y)
In Exercises 1-4, use the sum-to-product formulas to rewrite the sum or difference as a product. 1. sin 5Θ − sin 3Θ 2. sin 3Θ + sin Θ 3. cos 6x + cos 2x 4. cos x + cos 4x
In Exercises 1-4, use the sum-to-product formulas to find the exact value of the expression. 1. sin 75° + sin 15° 2. cos 120° + cos 60° 3. cos 3π / 4 - cos π / 4 4. sin 5π / 4 - sin 3π / 4
In Exercises 61-64, find all solutions of the equation in the interval [0,2π). Use a graphing utility to graph the equation and verify the solutions. 1. sin 6x + sin 2x = 0 2. cos 2x − cos 6x = 0
In Exercises 1-2, verify the identity. 1. csc 2Θ = csc Θ / 2 cos Θ 2. cos4x - sin4 x = cos 2x
In Exercises 1-4, solve the equation. 1. sin 2x − sin x = 0 2. sin 2x sin x = cos x 3. cos 2x − cos x = 0 4. cos 2x + sin x = 0
The Mach number M of a supersonic airplane is the ratio of its speed to the speed of sound. When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane.
The range of a projectile fired at an angle Θ with the horizontal and with an initial velocity of v0 feet per second is r = 1 / 32 vo2 sin 2 Θ where r is the horizontal distance (in feet) the
When two railroad tracks merge, the overlapping portions of the tracks are in the shapes of circular arcs (see figure). The radius r (in feet) of each arc and the angle Î˜ are related by x
The sine function is an odd function, so sin(-2x) = -2 sin x cos x.
sin u / 2 = - √1 - cos u / 2 When u is in the second quadrant.
Verify each identity for complementary angles ϕ and Θ. (a) sin(ϕ Θ) = cos 2Θ (b) cos(ϕ Θ) = sin 2Θ
Exercises 1-4, name the trigonometric function that is equivalent to the expression. 1. cos x / sin x 2. 1 / cos x 3. sin(π / 2 - x) 4. √cot2 x + 1
In Exercises 1 and 2, use the trigonometric substitution to write the algebraic expression as a trigonometric function of , where 0 < Θ < π / 2. 1. 25 - x2, x = 5 sin Θ 2. √x2 - 16, x = 4 sec

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