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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$.
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$?
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})$.
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$?
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$.
Prove that $coth^{-1}x = rac{1}{2} ext{log} rac{(x+1)}{(x-1)}$.
Prove that $coth^{-1}x = rac{1}{2} ext{log} rac{(x+1)}{(x-1)}$.
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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