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basic technical mathematics

Basic Technical Mathematics 12th Edition Allyn J. Washington, Richard Evans - Solutions

Sketch the appropriate graphs, and check each on a calculator. In a laser experiment, two mirrors move horizontally in equal and opposite distances from point A. The laser path from and to point B is shown in Fig. 10.30. From the figure, we see that x = a tanθ.Sketch the graph of x = f(θ) for a =
Graph the given functions.y = 8| cos(π/2x)|
Use a calculator to display the Lissajous figures defined by the given parametric equations.x = 4 cos 3t, y = cos 2t
The height (in ft) of a person on a ferris wheel above its center is given by h = −32 cos t where t is the time (in s). Graph one complete cycle of this function.
Sketch the appropriate graphs, and check each on a calculator. Near Antarctica, an iceberg with a vertical face 200 m high is seen from a small boat. At a distance x from the iceberg, the angle of elevation θ of the top of the iceberg can be found from the equation x = 200 cotθ. Sketch x as a
Sketch the appropriate graphs, and check each on a calculator. A mechanism with two springs is shown in Fig. 10.31, where point A is restricted to move horizontally. From the law of sines, we see that b = (a sin B) csc A. Sketch the graph of b as a function of A for a = 4.00 cm and B = π/4.
Write the equation for the given function with the given amplitude, period, and displacement, respectively.cosine, 12, 1/2, 1/8
Graph the given functions.y = 0.4|sin 6x|
Use a calculator to display the Lissajous figures defined by the given parametric equations.x = 2 sinπt, y = 3 sin 3πt
The horizontal displacement d (in m) of the bob on a large pendulum is d = 5 sin t, where t is the time (in s). Graph two cycles of this function.
Sketch the curves of the given trigonometric functions. Check each using a calculator.y = 3 cos(2πx + π)
Write the equation for the given function with the given amplitude, period, and displacement, respectively.sine, 18, 4, −1
Display the graphs of y = 2 sin 3x and y = 2 sin (−3x) on a calculator. What conclusion do you draw from the graphs?
Use a calculator to display the Lissajous figures defined by the given parametric equations. x = 3t 1+1³ y = 31² 1+1³
Use a calculator to display the Lissajous figures defined by the given parametric equations.x = t cos t, y = t sin t (t ≥ 0)
The graph displayed on an oscilloscope can be represented by y = −0.05 sin x. Display this curve on a calculator.
What conclusion do you draw from the calculator graphs of Y₁ = = 2sin (3x + 2) 6 and y₂ = -2sin[-(3x + 2)²
Sketch the appropriate graphs, and check each on a calculator. A cantilever column of length L will buckle if too large a downward force P is applied d units off center. The horizontal deflection x (see Fig. 10.32) is x = d [sec(kL) − 1], where k is a constant depending on P, and 0 < kL <
Sketch the curves of the given trigonometric functions. Check each using a calculator.y = 0.3tan 0.5x
Show that the given equations are identities. By viewing the graphs of y1 sin x and y2 = cos( x − π/2), show that cos( x − π/2) = sin x.
Display the graphs of y = 2 cos 3x and y = cos(−3x) on a calculator. What conclusion do you draw from the graphs?
The displacement y (in cm) of the end of a robot arm for welding is y = 4.75 cos t, where t is the time (in s). Display this curve on a calculator.
The graph of a function of the form y = a sin x or y = a cos x is shown. Determine the specific function of each. -4 0 12吋 X
Use a calculator to display the Lissajous figures defined by the given parametric equations. x = 2 sin 2t, y y = 2 sin³ 2t cost
Show that the given equations are identities. Show that cos(2x − 3π/8) = cos(3π/8 − 2x).
By noting the periods of sin 2x and sin 3x, find the period of the function y = sin 2x+sin 3x by finding the least common multiple of the individual periods.
The current in an alternating-current circuit is given by i = 12sin(120πt). Find a function for the voltage v if the amplitude is 18V and voltage lags current by 60°. Then find the displacement. See Example 5.Data from Example 5In alternating-current (ac) circuits, voltage (v) and current (i) can
Use a calculator to display the Lissajous figures defined by the given parametric equations.x = 2 cot t, y = 1 − cos 2t
The graph of a function of the form y = a sin x or y = a cos x is shown. Determine the specific function of each. y 0.2 0 -0.2 fi 2TT X
Sketch the curves of the given trigonometric functions. Check each using a calculator.y = −3 csc x
The graph of a function of the form y = a sin x or y = a cos x is shown. Determine the specific function of each. 1.5 0 -1.5 y 2п X
By noting the period of cos 1/2x and cos 1/3x find the period of the function y = cos 1/2x + cos 1/3x by finding the least common multiple of the individual periods.
Find the function and graph it for a function of the form y = −2 sin bx that passes through (π/4, −2) and for which b has the smallest possible positive value.
Parametric equations can be defined for functions other than trigonometric functions. In Exercises draw the graphs of the indicated parametric equations.x = √t + 4, y = 3√t − 1
Sketch the curves of the given functions by addition of ordinates.y = 2 + 1/2 sin 2x
Find the function and graph it for a function of the form y = 2 sin bx that passes through (π/6, 2) and for which b has the smallest possible positive value.
The current in an alternating-current circuit is given by i = 8.50sin(120πt ). Find a function for the voltage v if the amplitude is 28V and voltage leads current by 45°. Then find the displacement. See Example 5.Data from Example 5In alternating-current (ac) circuits, voltage (v) and current (i)
The graph of a function of the form y = a sin x or y = a cos x is shown. Determine the specific function of each. y 6 0 1 -6 2 x
A wave traveling in a string may be represented by the equationHere, A is the amplitude, t is the time the wave has traveled, x is the distance from the origin, T is the time required for the wave to travel one wavelength λ (the Greek letter lambda). Sketch three cycles of the wave for which A =
Parametric equations can be defined for functions other than trigonometric functions. In Exercises draw the graphs of the indicated parametric equations.x = t2 + 1, y = t − 1
Parametric equations can be defined for functions other than trigonometric functions, display the graphs on a calculator. x = √√/2t 1, y = 2t² - 3 -
Sketch the curves of the given functions by addition of ordinates.y = 1/2x − cos1/3 x
Use the fact that the frequency, in cycles/s (or Hz), is the reciprocal of the period (in s).Middle C on a piano has a frequency of 262 Hz. Find the time required for one complete cycle of the sound wave (the period) to three significant digits.
Parametric equations can be defined for functions other than trigonometric functions, display the graphs on a calculator.x = 4 − t, y = t3
Each ordered pair given in Exercises is located on the graph of either y = a sin x or y = a cos x where the amplitude is |a| = 2.50. Use the trace feature on a calculator to determine the equation of the function that contains the given point. (All points are located such that the x value is
Sketch the curves of the given functions by addition of ordinates.y = sin 2x + 3cos x
Use the fact that the frequency, in cycles/s (or Hz), is the reciprocal of the period (in s). The current in a certain alternating-current circuit is given by i = 2.5 sin(120πt). Find the period and the frequency.
The electric current i (in μA) in a certain circuit is given by i = 3.8cos2π(t+0.20), where t is the time in seconds. Sketch three cycles of this function.
Each ordered pair given in Exercises is located on the graph of either y = a sin x or y = a cos x where the amplitude is |a| = 2.50. Use the trace feature on a calculator to determine the equation of the function that contains the given point. (All points are located such that the x value is
Sketch the curves of the given functions by addition of ordinates.y = sin 3x + 2 cos 2x
The standard electric voltage in a 60-Hz alternating-current circuit is given by V = 170 sin 120πt, where t is the time in seconds. Sketch the graph of V as a function of t for 0 ≤ t ≤ 0.05 s.
A certain satellite circles Earth such that its distance y, in miles north or south (altitude is not considered) from the equator, is y = 4500 cos(0.025t − 0.25), where t is the time (in min) after launch. Use a calculator to graph two cycles of the curve.
Sketch the appropriate curves. A calculator may be used. An object oscillating on a spring has a displacement (in ft) given by y = 0.4 sin 4t + 0.3 cos 4t, where t is the time (in s). Sketch the graph.
Sketch the appropriate curves. A calculator may be used. An analysis of the temperature records of Louisville, Kentucky, indicates that the average daily temperature T (in °F ) during the year is approximatelywhere x is measured in months (x = 0.5 is Jan. 15, etc.). Sketch the graph of T vs. x for
Each ordered pair given in Exercises is located on the graph of either y = a sin x or y = a cos x where the amplitude is |a| = 2.50. Use the trace feature on a calculator to determine the equation of the function that contains the given point. (All points are located such that the x value is
Display the curves of the given functions on a calculator.y = 2 sin x − cos 2x
To tune the instruments of an orchestra before a concert, an A note is struck on a piano. The piano wire vibrates with a displacement y (in mm) given by y = 3.20 cos 880πt, where t is in seconds. Sketch the graph of y vs. t for 0 ≤ t ≤ 0.01 s.
Give the specific form of the equation by evaluating a, b, and c through an inspection of the given curve. Explain how a, b, and c are found.y = a sin(bx + c)(Fig. 10.21) 5 -5 0 Fig. 10.21 15 - X
In performing a test on a patient, a medical technician used an ultrasonic signal given by the equation I = Asin(ωt + ∅). Use a calculator to view two cycles of the graph of I vs. t if A = 5 nW/m2, ω = 2 × 105 rad/s, and ∅ = 0.4. Explain how you chose your calculator’s window settings.
Sketch the appropriate curves. A calculator may be used. The voltage e in a certain electric circuit is given by e = 50 sin 50πt + 80 sin 60πt, where t is the time (in s). Sketch the graph.
Display the curves of the given functions on a calculator. ㅠ cos(x + +4) y co = 0.4 sin 2x
Each ordered pair given in Exercises is located on the graph of either y = a sin x or y = a cos x where the amplitude is |a| = 2.50. Use the trace feature on a calculator to determine the equation of the function that contains the given point. (All points are located such that the x value is
The graph of a function of the form y = a sin bx or y = a cos bx is shown. Determine the specific function of each. 0.5 y 0 -0.5 - fir X T
Give the specific form of the equation by evaluating a, b, and c through an inspection of the given curve. Explain how a, b, and c are found.y = a cos(bx + c)(Fig. 10.21) y 5 -5 0 Fig. 10.21 15 X
Display the curves of the given functions on a calculator.y = 10 sin 3x − 20 cos x
Graph the indicated functions. The velocity v (in in./s) of a piston is v = 450 cos 3600t, where t is in seconds. Sketch the graph of v vs. t for 0 ≤ t ≤ 0.006 s.
Display the curves of the given functions on a calculator. y = sin.x X
Give the specific form of the equation by evaluating a, b, and c through an inspection of the given curve. Explain how a, b, and c are found.y = a cos(bx + c)(Fig. 10.22) 0.8 A Fig. 10.22 У -0.8 FI+ ST 4 X
Give the specific form of the equation by evaluating a, b, and c through an inspection of the given curve. Explain how a, b, and c are found.y = a sin(bx + c)(Fig. 10.22) 0.8 A Fig. 10.22 У -0.8 FI+ ST 4 X
On a Florida beach, the tides have water levels about 4 m between low and high tides. The period is about 12.5 h. Find a cosine function that describes these tides if high tide is at midnight of a given day. Sketch the graph.
Sketch the appropriate curves. A calculator may be used.An analysis of data shows that the mean density d (in mg/cm3) of a calcium compound in the bones of women is given by d = 139.3 + 48.6 sin(0.0674x − 0.210), where x represents the ages of women (20 ≤ x ≤ 80 years). (A woman is considered
Display the curves of the given functions on a calculator.y = 2 cosπx + cos(2πx − π)
The graph of a function of the form y = a sin bx or y = a cos bx is shown. Determine the specific function of each. 00 y -8
The graph of a function of the form y = a sin bx or y = a cos bx is shown. Determine the specific function of each. y 0 -4 2 X
The graph of a function of the form y = a sin bx or y = a cos bx is shown. Determine the specific function of each. 0.1 0 -0.1 y X<
Sketch the appropriate curves. A calculator may be used.A normal person with a pulse rate of 60 beats/min has a blood pressure of “120 over 80.” This means the pressure is oscillating between a high (systolic) of 120 mm of mercury (shown as mmHg) and a low (diastolic) of 80 mmHg. Assuming a
Give the specific form of the indicated equation by determining a, b, and c through an inspection of the given curve.y = a sin(bx + c)(Fig. 10.47) 2 料 4 -2 y 0 Fig. 10.47 3 4 一大
Sketch the appropriate curves. A calculator may be used.In the study of optics, light is said to be elliptically polarized if certain optic vibrations are out of phase. These may be represented by Lissajous figures. Determine the Lissajous figure for two light waves given by w1 = sinωt and W2 sin
Sketch the appropriate curves. A calculator may be used. The strain e (dimensionless) on a cable caused by vibration is e = 0.0080 − 0.0020 sin 30t + 0.0040 cos10t, where t is measured in seconds. Sketch two cycles of e as a function of t.
Display the curves of the given functions on a calculator.y = √x sin 0.5x
Display the curves of the given functions on a calculator.What conclusion can be drawn from the graph? y 1) =sin(x + 4) - cos(x-7)+1
Give the specific form of the indicated equation by determining a, b, and c through an inspection of the given curve.y = a cos(bx + c)(Fig. 10.47) 77 4 -2 y 0 Fig. 10.47 3m
Sketch the appropriate curves. A calculator may be used. The intensity I of an alarm (in dB—decibel) signal is given by I = 40 + 50 sin t − 20 cos 2t, where t is measured in seconds. Display two cycles of I as a function of t on a calculator.
Give the specific form of the indicated equation by determining a, b, and c through an inspection of the given curve.y = a cos(bx + c)(Fig. 10.48) y 凡 0 1 Fig. 10.48 9 X
Display the curves of the given functions on a calculator.y = sin2x + cos2x (sin2x = ( sin x)2)What conclusion can be drawn from the graph?
Sketch the appropriate curves. A calculator may be used.In checking electric circuit elements, a square wave such as that shown in Fig. 10.45 may be displayed on an oscilloscope. Display the graph of on a calculator, and compare it with Fig. 10.45. This equation gives the first three terms of
Sketch the appropriate curves. A calculator may be used. The available solar energy depends on the amount of sunlight, and the available time in a day for sunlight depends on the time of the year. An approximate correction factor (in min) to standard time is C = 10 sin 1/29 (n − 80) − 7.5 cos
Sketch the appropriate curves. A calculator may be used. Two signals are seen on an oscilloscope as being at right angles. The equations for the displacements of these signals are x = 4 cos πt and y = 2sin 3πt. Sketch the figure that appears on the oscilloscope.
Sketch the appropriate curves. A calculator may be used.Another type of display on an oscilloscope may be a sawtooth wave such as that shown in Fig. 10.46. Display the graph ofon a calculator and compare it with Fig. 10.46. See Exercise 49.Data from Exercises 49In checking electric circuit
Give the specific form of the indicated equation by determining a, b, and c through an inspection of the given curve.y = a sin(bx + c)(Fig. 10.48) 0 -1 y Fig. 10.48 x
Display the Lissajous figures on a calculator. sin(t +). x = sin 9 y = sint
Display the Lissajous figures on a calculator. x = = -4), 2 cos(2nt + y = COS πt
Display the Lissajous figures on a calculator. ㅠ x = cos(t - ), y cos(21+5) 3
Display the Lissajous figures on a calculator.x = 5cos t + 3, y = 9 sin t − 1
Display the Lissajous figures on a calculator.x = 10 cos3t, y = 10 sin3t
Display the Lissajous figures on a calculator.x = −cos 2πt, y = 2 sin πt
Display the function y = 2 |sin 0.2πx|−|cos 0.4πx| on a graphing calculator.
Display the function y = 0.2 |tan 2x| on a calculator
Show that cos( x + π/4) = sin(π/4 − x) on a calculator.
Sketch the appropriate curves. A calculator may be used. The range R of a rocket is given by Sketch R as a function of θ for v0 = 1000 m/s and g = 9.8 m/s 2. See Fig. 10.49. R = v sin 20 8

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