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mathematics
basic technical mathematics
Basic Technical Mathematics 12th Edition Allyn J. Washington, Richard Evans - Solutions
Simplify the given expressions. The result will be one of sin x, cos x, tan x, cot x, sec x, or csc x. secx - cosx tan.x
Prove the given identities.2csc2x cot x = 1 + cot2 x
Find an algebraic expression for each of the given expressions.sin(sec−1 x/4)
Find the angles of a triangle if one side is twice another side and the angles opposite these sides differ by 60°.
Explain how the exact value of sin 75° can be found using either Eq. (20.9) or Eq. (20.11).
A vertical pole of length L is placed on top of a hill of height h. From the plain below the angles of elevation of the top and bottom of the pole are α and β. See Fig. 20.12. Show thatFig. 20.12. h = Lcosasin 3 sin(a - 3)
Express sin 3x in terms of sin x only.
In electronics, in order to find the root-mean-square current in a circuit, it is necessary to express sin2 ωt in terms of cos2ωt. Show how this is done.
Prove the given identities.sin x cot2 x = csc x − sin x
Find an algebraic expression for each of the given expressions.cos(tan−1 x/3)
If two musical tones of frequencies 220 Hz and 223 Hz are played together, beats will be heard. This can be represented by y = sin 440πt + sin 446πt. Graph this function and estimate t (in s) when y = 0 between beats for 0.15 < t < 1.15 s.
Express cos 3x in terms of cos x only.
Prove the given identities. 1 sin ²0 1 - cos ²0 cot ²0
Prove the given identities. cos 20 cos²0 = 1 tan²0
In studying interference patterns of radio signals, the expression 2E2 − 2E2 cos(π − θ) arises. Show that this can be written as 4E2 cos2 (θ/2).
The index of refraction n, the angle A of a prism, and the minimum angle of deflection ∅ are related bySee Fig. 20.24. Show that an equivalent expression is n = sin (A + o) sin A
Find an algebraic expression for each of the given expressions.sec(csc−1 3x)
The acceleration due to gravity g (in m/s2) varies with latitude, approximately given by g = 9.7805(1 + 0.0053sin2θ), where θ is the latitude in degrees. Find θ for g = 9.8000.
For a first-quadrant angle, express the first function listed in terms of the second function listed.sin x, sec x
Under certain conditions, the electric current i (in A) in the circuit shown in Fig. 20.33 is given below. For what value of t (in s) is the current first equal to zero?i = −e−100t (32.0sin624.5t + 0.200 cos624.5t) - 20 Ω elle 0.10 Η Η 25 μF 3 Fig. 20.33
The design of a certain three-phase alternating-current generator uses the fact that the sum of the currents I cos(θ + 30°), I cos(θ + 150°), and I cos(θ + 270°) is zero. Verify this.
Use the half-angle formulas to solve the given problems. For the structure shown in Fig. 20.25, show that x = 2l sin21/2θ. F -1 Fig. 20.25
Express cos 4x in terms of cos x only.
Find an algebraic expression for each of the given expressions.tan(sin−1 2x)
For a first-quadrant angle, express the first function listed in terms of the second function listed.cos x, csc x
The current (in A) in a certain AC circuit is given by i = 4sin(120πt + π/2). Use the sum formula for sine to write this in a different form and then simplify.
Express sin 4x in terms of sin x and cos x.
In designing track for a railway system, the equation d = 4r sin2 A/4 is used. Solve for d in terms of cos A.
Verify each identity by comparing the graph of the left side with the graph of the right side on a calculator. X 2(sin + cos) 2 sin x sec+ csc =
Show that sin2 x(1 − sec2 x) + cos2 x(1 + sec4 x) has a constant value.
Solve for the angle A for the given triangles in the given figures in terms of the given sides and angles, explain your method. A C Fig. 20.42 a
Solve for the angle A for the given triangles in the given figures in terms of the given sides and angles, explain your method. A b Fig. 20.43 a B
Evaluate the given expressions.sin−1 x + cos−1 x
Show that cot y csc y sec y − csc y cos y cot y has a constant value.
Solve for x.y = 2 cos 2x
Solve for the angle A for the given triangles in the given figures in terms of the given sides and angles. A a b Fig. 20.44 B
If tan x + cot x = 2, evaluate tan2 x + cot2 x.
If sec x + cos x = 2, evaluate sec2 x + cos2 x.
Solve for x.y = π/4 − 3sin−1 5x
Solve for the angle A for the given triangles in the given figures in terms of the given sides and angles. A с B a Fig. 20.45
Use the given substitutions to show that the given equations are valid. In each, 0 < θ < π/2.If x = cosθ, show that √1-x² = sin 0.
Prove thatby expressing each function in terms of its x, y, and r definition. csc 0 tane + cote = cose
Use the given substitutions to show that the given equations are valid. In each, 0 < θ < π/2.If x = 3 sin θ, show that √9-x² 3 cos 0.
The height of the Statue of Liberty is 151 ft. See Fig. 20.46. From the deck of a boat at a horizontal distance d from the statue, the angles of elevation of the top of the statue and the top of its pedestal are α and β, respectively. Show that -|(151 + tan 3) d a = tan
Prove that sec2θ + csc2θ = sec2θ csc2θ by expressing each function in terms of its x, y, and r definition.
Solve for x.2y = sec−1 4x − 2
Use the given substitutions to show that the given equations are valid. In each, 0 < θ < π/2.If x = 2 tanθ, show that √4+x² 2 sec 0.
Solve the given equations for x (0 ≤ x < 2π).3(tan x− 2) = 1 + tan x
Solve the given equations for x (0 ≤ x < 2π).5 sin x = 3 − (sin x + 2)
Use the given substitutions to show that the given equations are valid. In each, 0 < θ < π/2.If x = 4secθ, show that √x² - 16 - 16 = 4 tan0.
Show that the length L of the pulley belt shown in Fig. 20.48 is L = 24 + 11 + 10sin-1.
Explain why sin−1 2x is not equal to 2sin−1 x.
A commemorative plaque is in the ground between two buildings that are 25 m and 50 m high, and are 35 m apart. See Fig. 20.47. Express the angle θ between the angles of elevation to the tops of the buildings from the plaque as a function of x, the distance from the taller building. Fig. 20.47 25
Solve the given equations for x (0 ≤ x < 2π).2(1 − 2sin2 x) = 1
If a TV camera is x m from the launch pad of a 50-m rocket that is y m above the ground, find an expression for θ, the angle subtended at the camera lens.
Solve the given equations for x (0 ≤ x < 2π).sec x = 2 tan2 x
Solve the given equations for x (0 ≤ x < 2π).2 sin2 θ + 3cosθ − 3 = 0
Solve the given equations for x (0 ≤ x < 2π).2 sin 2x + 1 = 0
Solve the given equations for x (0 ≤ x < 2π).sin x = sin x/2
Solve the given equations for x (0 ≤ x < 2π).cos 2x = sin(−x)
Solve the given equations for x (0 ≤ x < 2π).sin 2x = cos3x
Solve the given equations for x (0 ≤ x < 2π).cos 3x cos x + sin 3x sin x = 0
Solve the given equations for x (0 ≤ x < 2π).sin2 (x/2) − cos x + 1 = 0
Solve the given equations for x (0 ≤ x < 2π).sin x + cos x = 1
Determine whether the equality is an identity or a conditional equation. If it is an identity, prove it. If it is a conditional equation, solve it for 0 ≤ x < 2π.tan x + cot x = csc x sec x
Determine whether the equality is an identity or a conditional equation. If it is an identity, prove it. If it is a conditional equation, solve it for 0 ≤ x < 2π.tan x − sin2 x = cos2 x − sec2 x
Determine whether the equality is an identity or a conditional equation. If it is an identity, prove it. If it is a conditional equation, solve it for 0 ≤ x < 2π.sin x cos x − 1 = cos x − sin x
Determine whether the equality is an identity or a conditional equation. If it is an identity, prove it. If it is a conditional equation, solve it for 0 ≤ x < 2π.2 tan x = sin 2x sec2 x
Use the given substitutions to show that the equations are valid for 0 ≤ θ < π/2.If x = 2 cosθ, show that √4x² = 2 sin 0.
Solve the given equations graphically.x + ln x− 3 cos2 x = 2
Use the given substitutions to show that the equations are valid for 0 ≤ θ < π/2.If x = 2 sec θ, show that √x² 4 2 tan 0. =
Solve the given equations graphically.esin x − 2 = x cos2 x
Solve the given equations graphically.2 tan−1 x + x2 = 3
Use the given substitutions to show that the equations are valid for 0 ≤ θ < π/2.If x = cos θ, show that √1 - x² X tan 0.
Solve the given equations graphically.3 sin−1 x = 6 sin x+ 1
Find the exact value of the given expression for the triangle in Fig. 20.49.sin 2θ Fig. 20.49 10 4 3 دیا
Find an algebraic expression for each of the given expressions.tan(cot−1 x)
Find the exact value of the given expression for the triangle in Fig. 20.49.sec 2θ Fig. 20.49 10 4 3 دیا
Find an algebraic expression for each of the given expressions.sec(sin−1 x)
Find the exact value of the given expression for the triangle in Fig. 20.49.cos(θ/2) Fig. 20.49 10 4 3 دیا
Use the given substitutions to show that the equations are valid for 0 ≤ θ < π/2.If x = tanθ, show that X √1 + x² = sin 0.
Find an algebraic expression for each of the given expressions.cos(sin−1 x/5)
Find the exact value of the given expression for the triangle in Fig. 20.49.tan(θ/2) Fig. 20.49 10 4 3 دیا
Find an algebraic expression for each of the given expressions.tan(sec−1 x/2)
Find an algebraic expression for each of the given expressions.cos(sin−1 x + tan−1 y)
Find an algebraic expression for each of the given expressions.sin(2cos−1 x)
For the triangle in Fig. 20.50 find the expression for sin 2A. Fig. 20.50 A с b a
Show that y = Asin 2t + Bcos2t may be written as y = Csin(2t + a), where C= √√A²+ B² and tana = BIA.
Forces A and B act on a bolt such that A makes an angle θ with the x-axis and B makes an angle θ with the y-axis as shown in Fig. 20.51. The resultant R has components Rx = Acosθ − Bsinθ and Ry = Asinθ + Bcos θ. Using these components, show that R = √A²+ B².
Use the methods and formulas of this chapter to solve the given problems.Prove: (cosθ + j sinθ)2 = cos 2θ + j sin 2θ (j = √−1)
For the triangle in Fig. 20.50, find the expression for sin (A/2). Fig. 20.50 A C b a
Use the methods and formulas of this chapter to solve the given problems.Prove: (cosθ + j sinθ)3 = cos 3θ + j sin 3θ (j = √−1)
Evaluate exactly: 2(tan2 x + sin2 x − sec2 x) + cos2x
Evaluate exactly: sec4 θ − sec2 θ − tan4 θ − tan2 θ
Solve the inequality sin 2x > 2cos x for 0 ≤ x < 2π.
Show that (cos2α + sin2 α)sec2 α has a constant value.
In studying the interference of light waves, the identityis used. Prove this identity. sin+x sin+x -sinx = sinx+ sin 2x
In the study of chemical spectroscopy, the equationarises. Solve for θ. wt = sin 00-0 R
Express cos(A+ B + C) in terms of sin A, sin B, sin C, cos A, cos B, and cos C.
For a certain alternating-current generator, the expression I cosθ + I cos(θ + 2π/3) + I cos(θ + 4π/3) arises. Simplify this expression.
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