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mathematics
basic technical mathematics

Questions and Answers of Basic Technical Mathematics

Give verbal statements equivalent to the given inequalities involving the number x.x < 5 or x > 7
Solve the given inequalities by displaying the solutions on a calculator. See Example 6.Data from Example 6Display the solution to the inequalityon a calculator.On the calculator, set y1 = abs(x/2
Solve the given inequalities on a calculator such that the display is the graph of the solution.3x < 2x + 1 < x − 5
Draw a sketch of the graph of the region in which the points satisfy the given system of inequalities. y < 4-x y
Solve the given inequalities. Graph each solution.2s − 3 < s − 5 < 3s − 3
Give verbal statements equivalent to the given inequalities involving the number x.x < −10 or 10 ≤ x < 20
In Example 3, change the exponent from 6 to 5.Data from Example 3Using the binomial formula, expand (2x + 3)6. In using the binomial formula for (2x + 3)6 , we use 2x for a, 3 for b, and 6 for n.
In Example 2, change all − signs to + in the series.Data from Example 2Find the sum of the infinite geometric seriesHere, we see that a 4. 1 = We find r by dividing any term by the previous term,
In Example 3, change −119 to −88 and then find the common difference.Data from Example 3Find the common difference between successive terms of the arithmetic sequence for which the third term is
Solve the given inequalities. Graph each solution.0 < 1 − x ≤ 3 or −1 < 2x − 3 < 5
In Example 3, change the decimal form to 0.012012012 . . . .Data from Example 3Find the fraction that has its decimal form 0.121212 . . . . This decimal form can be considered as being 0.12 + 0.0012
The dosage of a certain medicine is 25 mL for each 10 lb of the patient’s weight. What is the dosage for a person weighing 56 kg?
Determine each of the following as being either true or false. If it is false, explain why.For an arithmetic sequence, if a1= 5, d = 3, and n = 10, then a10 = 35.
In Example 5, change 1000 to 500 and then find the sum.Data from Example 5Find the sum of the first 1000 positive integers. The first 1000 positive integers form a finite arithmetic sequence for
Write the first five terms of the sequence for which (a) a1 = 8 and d = −1/2; (b) a1 = 8 and r = −1/2.
In Example 7, change 0.97 to 0.98.Data from Example 7Approximate the value of 0.977 by use of the binomial series. We note that 0.97 = 1 − 0.03, which means 0.977 = [1 + (−0.03)]7 . Using four
In Example 4, change “seventh” to “tenth.”Data from Example 4Find the seventh term of the geometric sequence for which the second term is 3, the fourth term is 9, and r > 0.To get from the
In Example 6, change 1/2 to 1/3 and then find the sum.Data from Example 6Find the sum of the first seven terms of the geometric sequence for which the first term is 2 and the common ratio is 1/2. We
Determine each of the following as being either true or false. If it is false, explain why.For a geometric sequence, if a3 = 8, a5 = 2, then r = 1/2.
Find the sum of the first seven terms of the sequence − 6, 2, 2/3 .....
Determine each of the following as being either true or false. If it is false, explain why.The sum of the geometric series 6 − 2 +4/3 −. . . is 9/2.
Find the indicated quantity for an infinite geometric series.a1 = 4, r = 1/2, S = ?
Write the first five terms of the arithmetic sequence with the given values.a1= 4, d = 2
If 4, x + 6, and 3x + 2 are the first three terms of an arithmetic sequence, find the sum of the first 10 terms.
Write down the first five terms of the geometric sequence with the given values.a1 = 6400, r = 0.25
Expand and simplify the given expressions by use of the binomial formula.(t + 4)3
Determine each of the following as being either true or false. If it is false, explain why.Evaluating the first three terms, (x + y)10 = x10 + 10x9y + 45x8y2 + . . .
Write the first five terms of the arithmetic sequence with the given values.a1 = 6, d = −1/2
Find the indicated quantity for an infinite geometric series.a3 = 68, r = −1/3, S = ?
Find the first three terms of the expansion of √1 - 4x.
Find the fraction equal to the decimal 0.454545 . . . .
Write down the first five terms of the geometric sequence with the given values.a1 = 0.09, r = −3/2
Expand and simplify the given expressions by use of the binomial formula.(x − 2)3
Find the indicated quantity for an infinite geometric series. S = 4 + 2/2, r = 品 ILS || ?
Find the indicated term of each sequence.1, 6, 11, . . . (17th)
Find the indicated quantity for an infinite geometric series.a1 = 0.5, S = 0.625, r = ?
Write the first five terms of the arithmetic sequence with the given values.a1 = 2.5, a5 = −1.5
Write down the first five terms of the geometric sequence with the given values.a1 = 1/6, r = 3
Find the indicated term of each sequence.1, −3, −7, . . . (21st)
Write the first five terms of the arithmetic sequence with the given values.a2 = −2, a5 = 43
Expand and simply the expression (2x − y)5.
Write down the first five terms of the geometric sequence with the given values.a3 = −12, r = 2
Expand and simplify the given expressions by use of the binomial formula.(4x2 + 5)4
Expand and simplify the given expressions by use of the binomial formula.(6 + 0.1)5
Find the indicated term of each sequence.500, 100, 20, . . . (9th)
Find the sums of the given infinite geometric series.20 − 1 + 0.05 − . . .
Find the nth term of the arithmetic sequence with the given values.1, 4, 7, . . . ; n = 8
There are 12 seats in the first row around a semicircular stage. Each row behind the first has 4 more seats than the row in front of it. How many rows of seats are there if there is a total of 300
Simplify the given expressions. 4 tan 40 1tan²40
Simplify the given expressions by using one of the basic formulas of the chapter. Then use a calculator to verify the result by finding the value of the original expression and the value of the
Simplify the given expressions by using one of the basic formulas of the chapter. Then use a calculator to verify the result by finding the value of the original expression and the value of the
Prove the given identities.cotθ secθ = cscθ
Evaluate each expression by first changing the form. Verify each by use of a calculator.cos 250°cos70° + sin 250°sin 70°
Evaluate each expression by first changing the form. Verify each by use of a calculator. coscos singsin
Solve the given trigonometric equations analytically and by use of a calculator. Compare results. Use values of x for 0 ≤ x < 2π.2 sin x + 1 = 0
Evaluate exactly the given expressions if possible.tan[sin−1(2/3)]
Evaluate each expression by first changing the form. Verify each by use of a calculator. tan + tan 1- tantan
Prove the given identities.csc2x(1 − cos2x) = 1
Find the value of cos(α/2) if tan α = −0.2917(90° < α < 180°).
Simplify the given expressions.2cos2 1/2 x − 1
Solve the given trigonometric equations analytically and by use of a calculator. Compare results. Use values of x for 0 ≤ x < 2π.3 − 4 cos x = 7 − (2 − cos x)
Evaluate exactly the given expressions if possible.cos−1[cos(−π/4)]
Simplify each of the given expressions. Expansion of any term is not necessary; recognition of the proper form leads to the proper result.sin 2x cos3x + cos2x sin3x
Prove the given identities.secθ(1 − sin2θ) = cosθ
Find the value of sin(α/2) if cos α = 0.4706(270° < α < 360°).
Simplify the given expressions.4sin 1/2x cos 1/2x
Solve the given trigonometric equations analytically and by use of a calculator. Compare results. Use values of x for 0 ≤ x < 2π.7 sin x − 2 = 3(2 − sin x)
Evaluate exactly the given expressions if possible.tan−1[tan(2π/3)]
Simplify each of the given expressions. Expansion of any term is not necessary; recognition of the proper form leads to the proper result.4sin 7x cos7x
Simplify each of the given expressions. Expansion of any term is not necessary; recognition of the proper form leads to the proper result. tan x + tan 2x 1- tan x tan 2x
Prove the given identities.sin x(1 + cot2x) = csc x
Prove the given identities.sin(x + y) sin(x − y) = sin2 x − sin2 y
Simplify the given expressions.8sin2 2x − 4
Solve the given trigonometric equations analytically and by use of a calculator. Compare results. Use values of x for 0 ≤ x < 2π.4 − 3 csc2 x = 0
Evaluate exactly the given expressions if possible.cos[tan−1(−5)]
Prove the given identities.csc x(csc x − sin x) = cot2 x
Prove the given identities.cos(x + y) cos(x − y) = cos2 x − sin2 y
Simplify the given expressions.6cos 3x sin 3x
Solve the given trigonometric equations analytically and by use of a calculator. Compare results. Use values of x for 0 ≤ x < 2π.|sin x| = 1/2
Evaluate exactly the given expressions if possible.sec[cos−1(−0.5)]
Prove the given identities. csc x COSX tanx = cotx
Simplify each of the given expressions. Expansion of any term is not necessary; recognition of the proper form leads to the proper result.2 − 4sin2 6x
Prove the given identities.cosθ cotθ + sinθ = cscθ
Prove the given identities.cos(α + β) + cos(α − β) = 2 cosα cosβ
Derive an expression for tan(α/2) in terms of sin α and cos α.
Simplify the given expressions.sin6θ/cos3θ
Solve the given trigonometric equations analytically and by use of a calculator. Compare results. Use values of x for 0 ≤ x < 2π.sin 4x − cos2x = 0
Evaluate exactly the given expressions if possible.cos(2csc−11)
Simplify each of the given expressions. Expansion of any term is not necessary; recognition of the proper form leads to the proper result. 2 + 2 cos2x
Verify each identity by comparing the graph of the left side with the graph of the right side on a calculator. cos( + x) = V3cosx — sinx 2
Simplify each of the given expressions. Expansion of any term is not necessary; recognition of the proper form leads to the proper result.cos2 2x − sin2 2x
Prove the given identities.tan(90° + x) = −cot x [Explain why Eq. (20.13) cannot be used for this, but Eqs. (20.9) and (20.10) can be used.]
Prove the given identities. cot0sec ²0 1 tan 0 tan
Prove the given identities. sin 2 1 - cosa 2sin
Derive an expression for cot(α/2) in terms of sin α and cos α.
Simplify the given expressions.cos4 u − sin4 u
Solve the given trigonometric equations analytically and by use of a calculator. Compare results. Use values of x for 0 ≤ x < 2π.3 cos x − 4 cos2 x = 0
Evaluate exactly the given expressions if possible.sin(2 tan−1 2)
Simplify each of the given expressions. Expansion of any term is not necessary; recognition of the proper form leads to the proper result. √32 32 - 32 cos4x
Simplify the given expressions. sin 3x cos3x sin x COS.X

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