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Algebra And Trigonometry 10th Edition Ron Larson - Solutions

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, with t = 0 corresponding to 12:00 A.M. (a) Algebraically find the times at which the high and low tides
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 seconds.(a) Use a graphing utility to graph the heights of these pistons in the same viewing window for 0
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 (1.00), water (1.33), and glass (1.50). Triangular prisms are often used to measure the index of
(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 function. Use a graphing utility to rule out incorrectly rewritten functions. (c) Add a trigonometric
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 verify the graph of p shown below. b) Find the period of each sine component of p. Is p periodic?
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 of the sum of two angles.
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 horizontal distance x and the vertical distance y are measured in feet. Find a formula for the maximum
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 Î¸ / 2.(b) Write the area of the triangle as a function of Î¸. Determine the
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) Use a graphing utility to graph the model, where W = 185 and 0Â° (c) At what angle is the force
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 = csc x - sin x 3. 1 + sin Θ / cos Θ + cos Θ 1 + sin Θ = 2 sec Θ 4. cos Θ cot Θ / 1 - sin
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. cot3 t / csc t = cot(csc2 t 1)
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 u 5. csc (u) = ________ 6. sec (u) = __________
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 - tan x = sec x(csc x - 2 sin 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 + tan y / 1 - tan x tan y = cot x + cot y / cot x cot y - 1
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 whether the equation is an identity, and (c)confirm the results of parts(a) and (b) algebraically 1. (1
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 + 2 cos4 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 + 1
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˜) / sin Î˜, 0o (a) Verify that the expression for s is equal to h cot Î˜.(b)
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 because sin2(0) + cos2(0) = 1 and 1 + tan2(0) = 1.
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 Θ = sin Θ 4. 1 + tan Θ = sec Θ
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 sin2 β
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 = 7π / 6 + 2nπ and θ = 11π / 6 + 2nπ, where n is an integer. 3. The equation 2 tan2 x − 3 tan x
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 equation 2 cos x - 1 = 0.
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 intersect.Left side: y = cos x +1Right side: y = sin x(b) Graph the equation y = cos x + 1 − sin x and find
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 = 0
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 cos x = 0 5. cos x = x
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 = 5π / 12 4. 2 cos2 4x - 1 = 0 (a) x = π / 16 (b) x = 3π /16
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 = 0, [0, π] 3. 4 cos2 x − 2 sin x + 1 = 0, [- π / 2, π / 2] 4. 2 sec2 x + tan x − 6 = 0, [- π
(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 solutions are the x-coordinates of the maximum and minimum points of f. (Calculus is required to
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 behavior of the function as x †’ 0. (d) How many solutions does the equation sin x / x = 0 have
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 behavior of the function as x†’0. (d) How many solutions does the equation cos 1/x = 0 have
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 the displacement (in meters) and t is the time (in seconds). Find the times when the weight is at the
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 graphing utility to graph the displacement function for 0 ≤ t ≤ 10. Find the time beyond which the
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). Find Î¸ when the range r of a projectile is given by r = 1 / 32 v02 sin 2Î¸.
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 ............................................. Chicago, C 1 ......................................................... 31.0 2
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 ride begins When t = 0. (a) During the first 32 seconds of the ride, when will a person's seat on 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 largest inscribed rectangle. (b) Determine the values of x for which A ‰¥ 1.
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 - 13.70. Use a graphing utility to graph f and g in the same viewing window. Describe the result. (c)
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 inverse trigonometric function.
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 table. (c) Use the graphing utility to graph the functions f and g. (d) Use the table and the graphs to
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 / 12 = π / 6 - π / 4
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 60° sin 15° 4. cos 130° cos 40° − sin 130° sin 40°
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 130° sin 10° 4. sin 100° cos 40° − cos 100° sin 40°
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 − v) 3. tan(u + v) 4. csc(u − v)
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 − v) 4. cot(v − 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 + Θ) 4. cot(Θ π)
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 + π / 4) - sin(x - 7π / 6) = √3 / 2
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°) b. sin 135° − cos 30° 10. a. cos(120° + 45°) b. cos 120° + cos 45°
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 + π / 2) + cos2 x = 0 4. cos(x - π / 2) - sin2 x = 0
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 modeled by y = 1 / 3 sin 2t + 1 /4 cos 2t where y is the displacement (in feet) from equilibrium of the
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 wavelength Î». The models for two such waves areShow that
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 supplementary, sin α cos β = cos α sin β. 4. When A, B, and C form ∆ ABC, cos(A + B) = −cos
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), where C = arctan(b / a) and a > 0 4. a sin BΘ + b cos BΘ = √a2 + b2 cos(BΘ − C), where 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 2. 3 sin 2Θ + 4 cos 2Θ
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 the two lines. Then use your formula to find the angle between the given pair of lines.1. y = x and
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 2 2. y1 = sin(x + 4), y2 = sin x + sin 4
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 / 2 < u < 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 formulas. 1. cos u = 7 / 25, 0 < u < π / 2 2. sin u = 5 / 13, π / 2 < u < π 3. tan u = - 5 / 12, 3π / 2
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 - 1 = 0
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 Mach number is related to the apex angle Î˜ of the cone by sin(Î˜ 2) =
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 projectile travels. An athlete throws a javelin at 75 feet per second. At what angle must the athlete
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 / 2 = 2r sin2 Î˜ / 2. Write a formula for x in terms of cos Î˜.
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 Θ

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