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Questions and Answers
What is the solution to the equation $x^3 + 2x - 5 = 0$ using the Regula falsi method after three iterations?
What is the solution to the equation $x^3 + 2x - 5 = 0$ using the Regula falsi method after three iterations?
Approximately 1.5.
Show that all roots of $(x + 1)^6 + (x - 1)^6 = 0$ are given by $-i ext{cot} rac{(2k+1) au}{12}$ for $k = 0, 1, 2, 3, 4, 5$.
Show that all roots of $(x + 1)^6 + (x - 1)^6 = 0$ are given by $-i ext{cot} rac{(2k+1) au}{12}$ for $k = 0, 1, 2, 3, 4, 5$.
The roots are $-i ext{cot} rac{(2k + 1) au}{12}$ for integer $k$.
What is the cube root of unity and how does it relate to $(1 - w^6)=-27$?
What is the cube root of unity and how does it relate to $(1 - w^6)=-27$?
The cube root of unity is $w = e^{2 ext{π}i/3}$, and it satisfies $(1 - w^6) = -27$ due to the properties of roots of unity.
Express $ ext{sin}70^ ext{o}$ and $ ext{cos}70^ ext{o}$ in terms of sine and cosine of another angle $ heta$.
Express $ ext{sin}70^ ext{o}$ and $ ext{cos}70^ ext{o}$ in terms of sine and cosine of another angle $ heta$.
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How can you express $ ext{cos}8 heta$ in terms of cosines of multiples of $ heta$?
How can you express $ ext{cos}8 heta$ in terms of cosines of multiples of $ heta$?
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Prove that $ ext{cos}5 heta ext{sin}3 heta = rac{1}{27}( ext{sin}80^ ext{o} + 2 ext{sin}60^ ext{o} - 2 ext{sin}40^ ext{o} - 6 ext{sin}20^ ext{o})$.
Prove that $ ext{cos}5 heta ext{sin}3 heta = rac{1}{27}( ext{sin}80^ ext{o} + 2 ext{sin}60^ ext{o} - 2 ext{sin}40^ ext{o} - 6 ext{sin}20^ ext{o})$.
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What is the value of $a$, $b$, and $c$ in the expression $ ext{sin}6 heta = a ext{cos}5 heta ext{sin} + b ext{cos}3 heta + c ext{cos} heta ext{sin}5 heta$?
What is the value of $a$, $b$, and $c$ in the expression $ ext{sin}6 heta = a ext{cos}5 heta ext{sin} + b ext{cos}3 heta + c ext{cos} heta ext{sin}5 heta$?
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If $ ext{sinh}x = 2/3$, find the value of $ ext{cosh}2x$.
If $ ext{sinh}x = 2/3$, find the value of $ ext{cosh}2x$.
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Prove that $coth^{-1}x = rac{1}{2} ext{log} rac{(x+1)}{(x-1)}$.
Prove that $coth^{-1}x = rac{1}{2} ext{log} rac{(x+1)}{(x-1)}$.
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If $u + iv = ( rac{ ext{π}}{7} + i ext{√}(x^2 -1))$, prove that $(u^2 + v^2)^2 = 2(u^2 - v^2)$.
If $u + iv = ( rac{ ext{π}}{7} + i ext{√}(x^2 -1))$, prove that $(u^2 + v^2)^2 = 2(u^2 - v^2)$.
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Study Notes
Tutorial 1 - Applied Mathematics I
- Solve the equations:
- xº - x5 + x4 - 1 = 0
- x7 + 64x4 + x3 + 64 = 0
- x4 - x3 + x2 - x + 1 = 0
Tutorial 1 - Question 2
- Show that all roots of (x + 1)⁶ + (x - 1)⁶ = 0 are given by - i Cot ((2k+1)π/12) where k = 0, 1, 2, 3, 4, 5
Tutorial 1 - Question 3
- Find the cube root of unity.
- If w is the complex cube root of unity, then prove that (1 - w⁶) = -27
Tutorial 1 - Question 4
- Express Sin70 and Cos70 in terms of sine & cosθ
Tutorial 1 - Question 5
- Express cos8θ in a series of cosines of multiples of θ
Tutorial 1 - Question 6
- Prove that cos5θsin3θ = (1/27)(Sin80 + 2sin60 - 2sin40 - 6sin20)
Tutorial 1 - Question 7
- Prove that cos6θ - sin6θ = (1/sin2θ)(Cos60 + 15Cos20)
Tutorial 1 - Question 8
- Prove that sin6θ/sin2θ = 16cos⁴θ - 16cos²θ + 3
Tutorial 1 - Question 9
- Find the continued product of the roots of √3/2 + (i√3)/2
Tutorial 1 - Question 10
- Show that sin5θ = (sin5θ - 5sin30 + 10sine)/16
Tutorial 1 - Question 11
- If Sin6θ = acos5θsine + bcos3θ + ccosθsin5θ Find the value of a, b, c
Tutorial 1 - Question 12
- If sinθcos3θ = acosθ + bcos3θ + ccos5θ + dcos7θ then find a, b, c, d.
Tutorial 2 - Applied Mathematics I
- If tanhx = 2/3, find the value of x and then cosh2x
Tutorial 2 - Question 2
- Solve the equation 17 coshx + 18 sinhx = 1 for real values of x
Tutorial 2 - Question 3
- If coshβ = √(x²/y²) and cosasinhβ = 4xy/(x²+y²), show that:
- cosec (α - iβ) + cosec (α + iβ) = (4x)/(x²+y²)
- cosec (α - iβ) - cosec (α + iβ) = (4iy)/(x²+y²)
Tutorial 2 - Question 4
- If coshx = secθ, prove that:
- x = log (secθ + tanθ)
- tanh(x/2) = tan(θ/2)
Tutorial 2 - Question 5
- If u + iv = (π/7 + i√(x² -1), prove that (u² + v²)² = 2(u² - v²)
Tutorial 2 - Question 6
- Prove that cosh⁻¹(√1 + x²) = tanh⁻¹(x²)
Tutorial 2 - Question 7
- Separate into real and imaginary parts tan⁻¹(x + iy)
Tutorial 2 - Question 8
- Separate into real and imaginary parts of tan⁻¹(eiθ)
Tutorial 2 - Question 9
- Prove that coth⁻¹x = (1/2)log((x+1)/(x-1))
Tutorial 2 - Question 10
- Prove that sin[ilog(a+b)/(a²+b²)] = a + iβ, find a and β
Tutorial 2 - Question 11
- If (a+ib)x+iy = a + iβ, find a and β
Tutorial 2 - Question 12
- Show that for real values of a and b e^(2aicot⁻¹(b)/(bi⁻¹ - 1)⁻¹a)[(bi+1)/(bi-1)] = 1
Tutorial 6 - Applied Mathematics I
- Find the roots of the equation x³ + 2x - 5 = 0 by Regula falsi method (Take 3 iterations)
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Description
This quiz covers various problems in applied mathematics, focusing on equations, roots of unity, and trigonometric identities. Participants will solve complex equations and prove key mathematical identities related to sine and cosine. Perfect for students in an applied mathematics course looking to reinforce their understanding of these concepts.
