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mathematics
basic technical mathematics

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

In Example 4, change cos 1/2x to sin 1/2x and on the right of the equal sign change the + to −.Data from Example 4Solve the equation cos(x/2) = 1 + cos x (0 ≤ x < 2π).By using the half-angle formula for cos (x/2) and then squaring both sides of the resulting equation, this equation can be
In Example 2, change 2x to 3x and then simplify.Data from Example 2Simplify the expression cos2 2x − sin2 2x. Since this is the difference of the square of the cosine of an angle and the square of the sine of the same angle, it fits the right side of Eq. (20.22). Therefore, letting α = 2x, we
Use a calculator to check the indicated basic identities for the given angles.Eq. (20.4) for θ = 38°Data from Eq. (20.4) tan = sin cos
Determine each of the following as being either true or false. If it is false, explain it why.cos 2 2α = 1 − sin2 2 α
In Example 5, change 3/5 to 4/5 and then evaluate sin 2α.Data from Example 5If cos α = 3/5 for a fourth-quadrant angle, from Fig. 20.16(a) we see that sin α = −4/5. Thus,In Fig. 20.16(b), angle 2α is shown to be in the third-quadrant, verifying the sign of the result. (cos α = 3 5, α =
Use the half-angle formulas to evaluate the given functions.cos 15°
Determine the values of the given functions as indicated.Find sin105° by using 105° = 60° + 45°.
Find an algebraic expression for cos(sin−1 x).
In Example 7, change the + to −.Data from Example 7Solve the equation cos3x cos x + sin 3x sin x = 1(0 ≤ x < 2π). The left side of this equation is of the general form cos(A − x), where A = 3x. Therefore, cos3x cos x + sin 3x sin x = cos(3x − x) = cos 2xThe original equation becomes
In Example 8(a), change 0.5 to 1.Data from Example 8(a)Find cos(sin−1 0.5).Knowing that the values of inverse trigonometric functions are angles, we see that sin−1 0.5 is a first-quadrant angle. Thus, we find sin−1 0.5 = π/6. The problem is now to find cos(π/6). This is, of course, √3/2,
Use a calculator to check the indicated basic identities for the given angles.Eq. (20.5) for θ = 280°Data from Eq. (20.5) cot = cos sin
Determine each of the following as being either true or false. If it is false, explain it why. cos(α − β) = cosα cosβ − sinα sinβ
In Example 9, change sin to cos.Data from Example 9Find sin(tan−1 x).We know that tan−1 x is another way of stating “the angle whose tangent is x.” Thus, let us draw a right triangle (as in Fig. 20.38) and label one of the acute angles as θ, the side opposite θ as x, and the side adjacent
Use the half-angle formulas to evaluate the given functions.sin 22.5°
In Example 6, change the + in the denominator to − and then simplify the expression on the left.Data from Example 6Simplify the expression 2 1 + cos2x
Determine the values of the given functions as indicated.Find tan 75° by using 75° = 30° + 45°.
The electric current as a function of the time for a particular circuit is given by i = 8.00e−20t (1.73cos10.0t − sin10.0t). Find the time (in s) when the current is first zero.
Determine each of the following as being either true or false. If it is false, explain it why. sin- 2 cosa 2
Prove it that. tana + tan 3 tana - tan 3 sin(a + 3) sin(a - 3)
Use the half-angle formulas to evaluate the given functions.sin 105°
Determine the values of the given functions as indicated.Find cos15° by using 15° = 60° − 45°.
Determine each of the following as being either true or false. If it is false, explain it why. sin-1(-1) = Зп 2
Determine the values of the indicated functions in the given manner.Find sin 60° by using the functions of 30°.
Use a calculator to check the indicated basic identities for the given angles.Eq. (20.6) for θ = 4π/3Data from Eq. (20.6)sin2 θ + cos2 θ = 1
Write down the meaning of each of the given equations. See Example 1.y = tan−1 5xData from Example 1(a) y = cos−1 x is read as “y is the angle whose cosine is x.” In this case, x = cos y.(b) y = tan−1 2x is read as “y is the angle whose tangent is 2x.” In this case, 2x = tan y.(c) y =
Solve the given trigonometric equations analytically (using identities when necessary for exact values when possible) for values of x for 0 ≤ x < 2π.sin x − 1 = 0
Use the half-angle formulas to evaluate the given functions.cos 112.5°
Determine the values of the given functions as indicated.Find sin15° by using 15° = 225° − 210°.
Prove that cot2 x − cos2 x = cot2 x cos2 x.
Determine the values of the indicated functions in the given manner.Find sin 120° by using the functions of 60°.
Use a calculator to check the indicated basic identities for the given angles.Eq. (20.7) for θ = 5π/6Data from Eq. (20.7)1 + tan2 θ = sec2 θ
Write down the meaning of each of the given equations. See Example 1.y = csc−1 4xData from Example 1(a) y = cos−1 x is read as “y is the angle whose cosine is x.” In this case, x = cos y.(b) y = tan−1 2x is read as “y is the angle whose tangent is 2x.” In this case, 2x = tan y.(c) y =
Solve the given trigonometric equations analytically (using identities when necessary for exact values when possible) for values of x for 0 ≤ x < 2π.2 cos x + 1 = 0
Determine each of the following as being either true or false. If it is false, explain it why.For 0 ≤ θ < 2π, the solution of the equation 2cosθ + 1 = 0 is θ = 2π/3, 4π/3.
A solar furnace uses a parabolic reflector to direct the sun’s rays to a common focal point. The largest solar furnace in the world, located in Odeillo, France, has a parabolic reflector with a vertical cross section that measures 130 ft high and 17.9 ft deep. Find the focal length (the distance
Determine the type of curve from the given information.The diagonal brace in a rectangular metal frame is 3.0 cm longer than the length of one of the sides. Determine the type of curve represented by the equation relating the lengths of the sides of the frame.
For nonzero values of a, b, c, and d, show that (a) Lines ax + by + c = 0 and ax + by + d = 0 are parallel, (b) Lines ax + by + c = 0 and bx − ay + d = 0 are perpendicular.
The electric power P (in W) dissipated in a resistance R (in Ω) is given by P = Ri2, where i is the current (in A) in the resistor. Find the equation for the total power of 64 W dissipated in two resistors, with resistances 2.0 Ω and 8.0 Ω, respectively, and with currents i1 and i2,
Determine the equation of the hyperbola for which the difference in distances from (−6, 0) and (6,0) is (a) 4,(b) 8.
If the equation of the small right circle in Fig. 21.39 is x2 + y2 = a2 , what is the equation of the large circle? (Circles are tangent as shown with centers on the x-axis; the small circles are congruent.) Fig. 21.39. y X
Find the rectangular equation of each of the given polar equations, identify the curve that is represented by the equation.r sin(θ + π/6) = 3
What is the general form of the equation of a family of ellipses with foci on the y-axis if each passes through the origin?
Find the area of the square in Exercise 44.Data from Exercises 44(−5, 6), (0, 8), (−3, 1), and (2, 3) are the vertices of a square.
Find the polar equation of each of the given rectangular equations.2xy = 1
Solve the given problems: sketch or display the indicated curves.Using a calculator, determine what type of graph is displayed by r = 3sec2 (θ/2).
The entrance to a building is a parabolic arch 5.6 m high at the center and 7.4 m wide at the base. What equation represents the arch if the vertex is at the top of the arch?
One circular solar cell has a radius that is 2.0 in. less than the radius r of a second circular solar cell. Determine the type of curve represented by the equation relating the total area A of both cells and r.
The eccentricity e of an ellipse is defined as e = c/a. A cam in the shape of an ellipse can be described by the equation x2 + 9y2 = 81. Find the eccentricity of this elliptical cam.
Find the equation of the line with positive intercepts that passes through (3, 2) and forms with the axes a triangle of area 12.
Two persons, 1000 m apart, heard an explosion, one hearing it 4.0 s before the other. Explain why the location of the explosion can be on one of the points of a hyperbola.
Display the graph of r = 5 − 4 cosθ on a calculator, using(a) The polar curve mode, (b) The parametric curve mode. (See Example 6).Data from Example 6View the graph of r = 1 − 2cosθ on a calculator.Using the mode feature, a polar curve is displayed using the polar graph option or
If (a,3) is a point on the parabola y = x2 + 2x, what is a?
Find the rectangular equation of each of the given polar equations.r2 = sin2θ
Find the polar equation of each of the given rectangular equations.x2 + xy + y2 = 2
The cross section of the roof of a storage building shown in Fig. 21.80 is hyperbolic with the horizontal beam passing through the focus. Find the equation of the hyperbola such that its center is at the origin. 3.0 m 18 m Fig. 21.80
Determine whether the circle x2 − 6x + y2 − 7 = 0 crosses the x-axis.
The vertical cross section of a culvert under a road is elliptical. The culvert is 18 m wide and 12 m high. Find an equation to represent the perimeter of the culvert with the origin at road level and 2.0 m above the top of the culvert. See Fig. 21.89. 2 m 12 m road -18 m- Fig. 21.89
The sun is at the focus of a comet’s parabolic orbit. When the comet is 4 ×107 km from the sun, the angle between the axis of the parabola and the line between the sun and comet is 60°. What is the closest distance the comet comes to the sun if p > 0?
A supersonic jet creates a conical shock wave behind it. What type of curve is outlined on the surface of a lake by the shock wave if the jet is flying horizontally?
The equation 4x − 2y = k defines a family of lines, one for each value of k. On a calculator display the lines for k = −4, k = 0, and k = 4. What conclusion do you draw about this family of lines?
Find the rectangular equation of each of the given polar equations. r = 3 sin0 + 4 cos 0
What is the length of the horizontal bar across the parabolically shaped window shown in Fig. 21.53? Bar. I 1 2.50 ft 12.50 ft 1 4.20 ft Fig. 21.53
A space object (dubbed 2003 UB313), larger and more distant than Pluto, was discovered in 2003. In its elliptical orbit with the sun at one focus, it is 3.5 billion miles from the sun at the closest, and 9.0 billion miles at the farthest. What is the eccentricity of its orbit? See Exercise 46.Data
In Fig. 21.95, if the plane cutting the cones passes through the intersection of the upper and lower cones, what type of curve is the intersection of the plane and cones? Fig. 21.95 --++- V Ellipse Circle Hyperbola Parabola
Display the graph of r = 4sin 3θ on a calculator, using (a) The polar curve mode, (b) The parametric curve mode. (See Example 6).Data from Example 6View the graph of r = 1 − 2cosθ on a calculator.Using the mode feature, a polar curve is displayed using the polar graph option or
Find the perimeter of the parallelogram in Exercise 43.Data from Exercises 43(−5,−4), (7, 1), (10, 5), and (−2, 0) are the vertices of a parallelogram.
Find the polar equation of each of the given rectangular equations.x2 + (y + 3)2 = 16
Find the points of intersection of the circle x2 + y2 − x − 3y = 0 and the line y = x − 1.
A plane is flying at a constant altitude of 2000 m. Show that the equation relating the horizontal distance x and the direct-line distance l from a control tower to the plane is that of a hyperbola. Sketch the graph of l as a function of x. See Fig. 21.81. 2000 m X 1 Fig. 21.81 Tower
Use the following definition to find the midpoints between the given points on a straight line. The midpoint between points (x1 , y1) and (x2 , y2) on a straight line is the point.(−4, 9) and (6, 1) (x₁ + x₂ ³₁ +3₂²) 2 2
Show that the determinant equation at the right defines a straight line. x b 10 10 1 y m = 0
The equation kx − 2y = 4 defines a family of lines, one for each value of k. On a calculator display the lines for k = −4, k = 0, and k = 4. What conclusion do you draw about this family of lines?
Halley’s Comet has an elliptical orbit with a = 17.94 AU (AU is astronomical unit, 1AU = 9.3 × 107mi ) and b = 4.552 AU, with the sun at one focus. What is the closest that the comet comes to the sun? Explain your method.
An electric current (in A) is i = 2 + sin(2πr − π/3). What is the equation for the current if the origin of the (t', i') system is taken as (1/6, 2) of the (t, i) system?
Find the rectangular equation of each of the given polar equations.r = 2sin 2θ
When graphed on a calculator, a circle sometimes looks like it is longer in one direction than it is in the other. Explain why this can happen.
The primary mirror in the Hubble space telescope has a parabolic cross section, which is shown in Fig. 21.54. What is the focal length (vertex to focus) of the mirror?
An architect designs a patio shaped such that it can be described as the area within the polar curve r = 4.0 − sinθ, where measurements are in meters. Sketch the curve that represents the perimeter of the patio.
A draftsman draws a series of triangles with a base from (−3, 0) to (3, 0) and a perimeter of 14 cm (all measurements in centimeters). Find the equation of the curve on which all of the third vertices of the triangles are located.
The radiation pattern of a certain television transmitting antenna can be represented by r = 120(1 + cosθ), where distances (in km) are measured from the antenna. Sketch the radiation pattern.
Find the rectangular equation of each of the given polar equations.r cosθ = 4 tanθ
The vertical cross section of the cooling tower of a nuclear power plant is hyperbolic, as shown in Fig. 21.125. Find the radius r of the smallest circular horizontal cross section.Fig. 21.125. 40 ft 100 ft 80 ft 1 100 ft
A satellite at an altitude proper to make one revolution per day around the center of Earth will have for an excellent approximation of its projection on Earth of its path the curvewhere R is the radius of Earth. Sketch the path of the projection. r2 = R² cos 2(0+1),
A jet travels 600 km at a speed of v km/h for t hours. Graph the equation relating v as a function of t.
Under certain load conditions, a beam fixed at both ends is approximately parabolic in shape. If a beam is 4.0 m long and the deflection in the middle is 2.0 cm, find an equation to represent the shape of the beam. See Fig. 21.55. 2.0 cm -4.0 m- Fig. 21.55
The voltage V across part of an electric circuit is given by V = E − iR, where E is a battery voltage, i is the current, and R is the resistance. If E = 6.00 V and V = 4.35 V for i = 9.17 mA, find V as a function of i. Sketch the graph (i and V may be negative).
A light beam is reflected off the edge of an optic fiber at an angle of 0.0032°. The diameter of the fiber is 48 μm. Find the equation of the reflected beam with the x-axis (at the center of the fiber) and the y-axis as shown in Fig. 21.27. y 0.0032° Fig. 21.27 D- X
An airplane wing is designed such that a certain cross section is an ellipse 8.40 ft wide (horizontally) and 1.20 ft thick (vertically). Find the equation of this ellipse if the center is at the origin.
A spotlight with a parabolic reflector is 15.0 cm wide and is 6.50 cm deep. See Fig. 21.56 and Example 4. Where should the filament of the bulb be located so as to produce a beam of light? Fig. 21.56 6.50 cm Filament 15.0 cm
A radio signal is sent simultaneously from stations A and B 600 km apart on the Carolina coast. A ship receives the signal from A 1.20 ms before it receives the signal from B. Given that radio signals travel at 300 km/ms, draw a graph showing the possible locations of the ship. This problem
Determine the number of real solutions of the given systems of equations by sketching the indicated curves.x² + y² − 4y − 5 = 0y² − 4x² − 4 = 0
Use the following definition to find the midpoints between the given points on a straight line. The midpoint between points (x1 , y1) and (x2 , y2) on a straight line is the point.(2.6, 5.3) and (−4.2,−2.7) (x₁ + x₂ ³₁ +3₂²) 2 2
Find the distance between the points (3, π/6) and (4, π/2) by using the law of cosines.
Use a calculator to view the circle 2x2 + 2y2 + 2y − x −1 = 0.
On a computer drawing showing the specifications for a mounting bracket, holes are to be drilled at the points (32.5, 25.5) and (88.0, 62.5), where all measurements are in mm. Find the distance between the centers of the two holes.
View the curves of the given equations on a calculator.x2 − 4y2 + 4x + 24y − 48 = 0
The grade of a highway is its slope expressed as a percent (a 5% grade means the slope is 5/100). If the grade of a certain highway is 6%, find(a) Its angle of inclination and (b) The change in elevation (in ft) of a car driving for 2.00 mi uphill along this highway (1mi = 5280 ft).
Use the following definition to find the midpoints between the given points on a straight line. The midpoint between points (x1 , y1) and (x2 , y2) on a straight line is the point.(−1, 6) and (−13,−8) (x₁ + x₂ ³₁ +3₂²) 2 2
Solve the given problems. All coordinates given are polar coordinates.Is the point (1/2, 3π/2) on the curve r = sin(θ/3)?
Use the following definition to find the midpoints between the given points on a straight line. The midpoint between points (x1 , y1) and (x2 , y2) on a straight line is the point.(−12.4, 25.7) and (6.8,−17.3) (x₁ + x₂ ³₁ +3₂²) 2 2

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