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mathematics
basic technical mathematics

Questions and Answers of Basic Technical Mathematics

Identify each of the equations as representing either a circle, a parabola, an ellipse, a hyperbola, or none of these.x(x − 3) = y(1 − 2y2)
Simplify the given expressions by giving the results in terms of one-half the given angle. Then use a calculator to verify the result. 1 + cos 96° 2
Find tan 184° directly and by using functions of 92°.
Write down the meaning of each of the given equations. See Example 1.y = 4 sec−1(3x + 2)Data from Example 1(a) y = cos−1 x is read as “y is the angle whose cosine is x.” In this case, x = cos
Simplify the given expressions by giving the results in terms of one-half the given angle. Then use a calculator to verify the result. 1- cos 328° 2
Multiply and simplify.(csc x − 1)(csc x + 1)
Solve the given trigonometric equations analytically (using identities when necessary for exact values when possible) for values of x for 0 ≤ x < 2π.3 tan x− cot x = 0
Find tan 5π/4 by using 5π/4 = π + π/4
Simplify the given expressions.sin x cos2x + sin 2x cos x
Find cos 96° directly and by using functions of 48°.
Multiply and simplify.tan u(cot u + tan u)
Solve the given trigonometric equations analytically (using identities when necessary for exact values when possible) for values of x for 0 ≤ x < 2π.2 sin2 x − sin x = 0
Evaluate exactly the given expressions if possible.cos−1 0.5
Simplify the given expressions. - cos 8x 2
Find cos π by using π = 2(π/2)
Simplify the given expressions.sin3x cos x − sin x cos3x
Find cos 276° directly and by using functions of 138°.
Solve the given trigonometric equations analytically (using identities when necessary for exact values when possible) for values of x for 0 ≤ x < 2π.sin 4x − sin 2x = 0
Find sin 3π/2 by using 3π/2 = 2(3π/4)
Simplify the given expressions. tan(x - y) + tanx 1 tan(x - y) tan y
Evaluate exactly the given expressions if possible.sin−11
Simplify the given expressions. 4 + 4 cos83 8
Multiply and simplify.cos2 t(1 + tan2 t)
Simplify the given expressions.cos5x cos x + sin 5x sin x
Find tan 2π/7 directly and by using functions of π/7.
Simplify the given expressions. √2 - 2 cos64x
Solve the given trigonometric equations analytically (using identities when necessary for exact values when possible) for values of x for 0 ≤ x < 2π.sin 2x sin x + cos x = 0
Find tan 60° by using 60° = 2(30°).
Evaluate exactly the given expressions if possible.tan−11
Factor and simplify.sin x + sin x tan2 x
Simplify the given expressions. The result will be one of sin x, cos x, tan x, cot x, sec x, or csc x. 1 + cos2x 2cosx
Use a calculator to verify the values found by using the double-angle formulas.Find sin 1.2π directly and by using functions of 0.6.
The time t as a function of the displacement d of a piston is given bySolve for d. 2πf cos-1. A
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 − sin x/2 = 0
Determine the values of the indicated functions in the given manner.Find cos 45° by using 45° = 1/2 (90°).
The path of a bouncing ball is given byShow that this path can also be shown asUse a calculator to show that this can also be shown as y = |sin x + cos x|. y = √(sinx + cos x)².
The equation for the trajectory of a missile fired into the air at an angle α with velocity v0 isHere, g is the acceleration due to gravity. On the right of the equal sign, combine terms and
Simplify the given expressions. The result will be one of sin x, cos x, tan x, cot x, sec x, or csc x. sin 2xsecx 2
Derive an expression for csc(α/2) in terms of sec α
Derive an expression for sec(α/2) in terms of sec α
Solve the given equations graphically.3 sin x − x = 0
Use a calculator to determine whether the given equations are identities. 2 cos²x - 1 sinxcos.x = tan x cotx
Use a calculator to determine whether the given equations are identities.secθ tanθ cscθ = tan2θ + 1
The CN Tower in Toronto is 553 m high, and has an observation deck at the 335-m level. How far from the top of the tower must a 553-m high helicopter be so that the angle subtended at the helicopter
Solve the given equations graphically.4 cos x + 3x = 0
Prove that the given expressions are equal. In exercise use the relation for sin(α + β) and show that the sine of the sum of the angles on the left equals the sine of the angle on the right. sin- +
Prove that the given expressions are equal, use the relation for tan(α + β) . tan 1 3 + tan 2 || π 4
Use a calculator to determine whether the given equations are identities.sin x cos x tan x = cos2 x − 1
Solve the given equations graphically.2 sin 2x = x2 + 1
Verify each identity by comparing the graph of the left side with the graph of the right side on a calculator. cos - sin cose + sin cot 0 1 cot + 1
Simplify the given expressions. The result will be one of sin x, cos x, tan x, cot x, sec x, or csc x.(sec x+ tan x)(1 − sin x)
To find the horizontal range R of a projectile, the equation R = vt cos α is used, where α is the angle between the line of fire and the horizontal, v is the initial velocity of the projectile, and
Solve the given equations graphically.√x − sin 3x = 1
Use a calculator to determine whether the given equations are identities.cos3 x csc3 x tan3 x = csc2 x − cot2 x
Evaluate the given expressions. sin(sin-11+ 2 + cos 5/
The cross section of a radio-wave reflector is defined by x = cos2θ, y = sinθ. Find the relation between x and y by eliminating θ.
Solve the given equations graphically.2 ln x = 1 − cos 2x
In determining the path of least time between two points under certain conditions, it is necessary to show thatShow this by transforming the left-hand side. 1 + cose 1- cose sin0 = 1+ cos
Verify each identity by comparing the graph of the left side with the graph of the right side on a calculator.sin 3y cos 2y − cos3y sin 2y = sin y
Solve the given equations graphically.ex = 1 + sin x
Show that the length l of the straight brace shown in Fig. 20.4 can be found from the equationFig. 20.4. 1= = a(1 + tano) sin
Verify each identity by comparing the graph of the left side with the graph of the right side on a calculator. sin 4x(cos² 2x - sin²2x) sin 8x 2
The path of a point on the circumference of a circle, such as a point on the rim of a bicycle wheel as it rolls along, tracks out a curve called a cycloid. See Fig. 20.5. To find the distance through
In analyzing light reflection from a cylinder onto a flat surface, the expression 3cosθ − cos3θ arises. Show that this equals 2cosθ cos2θ + 4sinθ sin2θ.
In the study of the stress at a point in a bar, the equation s = a cos2 θ + bsin2 θ − 2t sinθ cosθ arises. Show that this equation can be written as s = (a + b) + (a - b) cos 20 tsin 20. S
In finding the frequencies of vibration of a vibrating wire, the equation x tan x = 2.00 occurs. Find x if 0 < x < π/2.
Find an equivalent algebraic expression.sin(sin−1 x + cos−1 y)
Verify each identity by comparing the graph of the left side with the graph of the right side on a calculator. - sin = (1-2 sin)(1+ sin) cosx— sin
Verify each identity by comparing the graph of the left side with the graph of the right side on a calculator. sin x cscx— cotx = 1 + cos.x
The instantaneous electric power p in an inductor is given by the equation p = vi sinωt sin(ωt − π/2). Show that this equation can be written as p = −1/2 vi sin2ωt.
Verify each identity by comparing the graph of the left side with the graph of the right side on a calculator.csc 2x + cot 2x = cot x
An equation used in astronomy is θ − esinθ = M. Solve for θ for e = 0.25 and M = 0.75.
Find an equivalent algebraic expression.cos(sin−1 x − cos−1 y)
In Example 1, change the 9 to 36; find the vertices, ends of the minor axis, and foci; and sketch the ellipse.Data from Example 1The ellipseseems to fit the form of either Eq. (21.17) or Eq.
In Example 1, change the − before the 2y2 to + and then determine what type of curve is represented.Data from Example 1The equation 2x2 = 3 − 2y2 represents a circle. This can be seen by putting
Verify each identity by comparing the graph of the left side with the graph of the right side on a calculator. α tan
Evaluate the given expressions.sin−1 0.5 + cos−1 0.5
Explain how to transform sinθ tanθ + cosθ into secθ.
Evaluate the given expressions.tan−1√3 + cot−1√3
In Example 1(b), change π/6 to 5π/6.Data from Example 1(b)The graph of θ = π/6 is a straight line through the pole. It represents all points for which θ = π/6 for all possible values of r. This
Explain how to transform tan2θ cos2θ + cot2θ sin2θ into 1.
In Example 2, change (−2,−5) to (−2, 5).Data from Example 2The distance between (3,−1) and (−2,−5) is given by[It makes no difference which point is chosen as (x1 , y1) and which is
Evaluate the given expressions.sin−1 x + sin−1(−x)
A person at a baseball game looks at the scoreboard which is h m high. If the scoreboard is at a horizontal distance x, write an expression for the angle θ that the height of the scoreboard makes at
For both of the points plotted in Example 2, change 2π/3 to π/3 and then find another set of coordinates for each point, similar to those shown for the points in the example.Data from Example 2The
In Example 2, interchange the 4 and 9; find the vertices, ends of the minor axis, and foci; and sketch the ellipse.Data from Example 2The equation of the ellipsehas the larger denominator, 9, under
In Example 1, change 12x to 20x.Data from Example 1Find the coordinates of the focus and the equation of the directrix and sketch the graph of the parabola y2 = 12x. Because the equation of this
In Example 3, change y2 − 14 to 14 − y2 and then determine what type of curve is represented.Data from Example 3Identify the curve represented by 2x2 + 12x = y2 − 14. Determine the appropriate
(a) Find the distance between (4,−1) and (6, 3). (b) Find the slope of the line perpendicular to the line segment joining the points in part (a).
In Example 2, change y + 2 to y − 2 and change the sign before the second term from + to −. Then describe and sketch the curve.Data from Example 2Describe the curve of the equationWe see that
In Example 2, interchange the denominators of 4 and 16; find the vertices, ends of the conjugate axis, and foci. Sketch the curve.Data from Example 2The hyperbolahas vertices at (0,−2) and (0, 2).
Transform the given equations by rotating the axes through the given angle. Identify and sketch each curve.x2 − y2 = 25, θ = 45°
In Example 2, change (2,−1) to (−2, 1).Data from Example 2Find the equation of the line through (2,−1) and (6, 2). We first find the slope of the line through these points:Then by using either
Determine each of the following as being either true or false. If it is false, explain why.The distance between (4,−3) and (3,−4) is √2.
In Example 3, change (3,−5) to (−3,−5).Data from Example 3The slope of a line through (3,−5) and (−2,−6) isSee Fig. 21.5. Again, we may interpret either of the points as (x1, y1) and the
Identify the type of curve represented by the equation 2(x2 + x) = 1 − y2.
Transform the given equations by rotating the axes through the given angle. Identify and sketch each curve.x2 + y2 = 16, θ = 60°
Determine each of the following as being either true or false. If it is false, explain why.2y − 3x = 6 is a straight line with intercepts (2, 0) and (0, 3).
In Example 2, change −8x to −20x.Data from Example 2If the focus is to the left of the origin, with the directrix an equal distance to the right, the coefficient of the x-term is negative. This
In Example 3, change the fourth and fifth terms from −4y − 4 to +6y − 9 and then find the center.Data from Example 3Find the center of the hyperbola 2x2 − y2 − 4x − 4y − 4 = 0. To

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