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mathematics
basic technical mathematics

Questions and Answers of Basic Technical Mathematics

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.
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
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
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
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
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 −
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 −
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
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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