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Sunday, January 31, 2016

A parallel plate capacitor, dielectric constants

Question:
There is a parallel plate capacitor of area L^2 and separation distance of d. A half of the space, d/2, is filled with a material of dielectric constant, \kappa_1. The other half is filled with the other material of constant, \kappa_2. What is the capacitance of this capacitor if the free space capacitance is C_0?

Answer:
The capacitance of a parallel plate capacitor is
C = \frac{\epsilon_0\kappa A}{d}
where A=L^2. When a capacitor does not have any dielectrics (free space between plates), it will be
C_0 = \frac{\epsilon_0 A}{d}
We can consider the particular capacitor is like series connection of two different capacitors. Namely,
\frac{1}{C_{\mathrm{Total}}} = \frac{1}{C_1}+\frac{1}{C_2} \\ \rightarrow C_{\mathrm{Total}} = \frac{C_1C_2}{C_1+C_2}
From the condition, we have each capacitance of C_1 and C_2.
C_{1,2} =  \frac{\epsilon_0\kappa_{1,2} A}{\frac{1}{2}d} = \frac{2\epsilon_0\kappa_{1,2} A}{d}
Then, we can rewrite it in terms of C_0.
C_{1,2} = 2\kappa_{1,2}C_0

Plug in the above.
 C_{\mathrm{Total}} = \frac{2\kappa_1C_0\times 2\kappa_2 C_0}{2\kappa_1C_0+2\kappa_2 C_0}    \\       = \frac{4\kappa_1\kappa_2 C^2_0}{2C_0(\kappa_1+\kappa_2)}   \\       =  \frac{2\kappa_1\kappa_2 C_0}{\kappa_1+\kappa_2}  

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Saturday, January 30, 2016

De Moivre's theorem and application

Question:
By using De Moivre's theorem, simplify the following expressions:
(1) \left(\frac{1+\sqrt{3}i}{2}\right)^{10}
(2) \left(\frac{\sqrt{3}+i}{1+i}\right)^{6}

Answer:
De Moivre's theorem shows
(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta 
(1) First, we obtain the magnitude.
\left| \frac{1+\sqrt{3}i}{2} \right| = \sqrt{\left(\frac{1}{2}\right)^2+ \left(\frac{\sqrt{3}}{2}\right)^2} = 1
The argument \theta ranges from 0 to 360 degrees. Comparing with \cos\theta + i\sin\theta, we can have
\cos\theta = \frac{1}{2}, \sin\theta = \frac{\sqrt{3}}{2}
Therefore, \theta = 60^o. Use the theorem.
(\cos 60^o + i\sin 60^o)^{10} = \cos(10\times60^o)+i\sin(10\times60^o)   \\                      = \cos 600^o + i\sin 600^o   \\                      = \cos 240^o + i\sin 240^o    \\                      = -\frac{1}{2}-\frac{\sqrt{3}}{2}i  

(2)
Let us separate the numerator and denominator.
\frac{(\sqrt{3}+i)^6}{(1+i)^6}
For the numerator, the magnitude and argument are
|\sqrt{3}+i| = 2   \\     \cos\theta_1 = \frac{\sqrt{3}}{2}, \sin\theta_1 = \frac{1}{2} \rightarrow \theta_1 = 30^o  
Thus, \sqrt{3}+i = 2(\cos 30^o + i\sin 30^o).
For the denominator, we have
|1+i| = \sqrt{2}   \\     \cos\theta_2 = \frac{1}{\sqrt{2}}, \sin\theta_2 = \frac{1}{\sqrt{2}} \rightarrow \theta_2 = 45^o  
Thus, 1+i = \sqrt{2}(\cos 45^o + i\sin 45^o).
We have
\frac{\sqrt{3}+i}{1+i}=\frac{2(\cos 30^o + i\sin 30^o)}{\sqrt{2}(\cos 45^o + i\sin 45^o)} \\     = \sqrt{2}[\cos(-15^o)+i\sin(-15^o)]   
Use the theorem.
(\sqrt{2}[\cos(-15^o)+i\sin(-15^o)])^6 = 2^3[\cos(-6\times 15^o) + i\sin(-6\times 15^o)]  \\                = 8[\cos(-90^o) + i\sin(-90^o)]  \\                =-8i

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Dropping a coin and finding out the depth

Question:
You released a coin in order to find the depth of a well. The time between dropping the coin and hearing it hit the bottom is 2.059 s. The speed of sound is v_s=343 m/s. Calculate the depth, d, of the well.

Answer:
The total time T is divided into the time hitting the bottom, t_1 and time the sound traveling, t_2. Namely,
T=t_1+t_2
We have
d = \frac{1}{2}gt_1^2 \\   t_2 = \frac{d}{v_s}
Since t_1 = T - t_2, the depth can be expressed as
d = \frac{1}{2}g(T-\frac{d}{v_s})^2
Expand and arrange it in terms of d.
d^2 - d(\frac{2v^2}{g}+2v_sT)+v_s^2T^2=0
Plug v_s=343 m/s and g=9.8 m/s^2 into the above.
d^2 -25,422.5d + 498,770.7 = 0 
The solution is
d=\frac{-(25,422.5)\pm\sqrt{(25,422.5)^2-4\times 498,770.7}}{2} \\    = 19.6 \mathrm{m}
The other solution is neglected.

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Equilibrium of forces

Question:
When this system is in equilibrium, find the tension in cord b, T_b. The hanging mass is 2.00 kg.

Answer:
As shown in the second figure, the tensions are vectors on x-y coordinate. Since these are at equilibrium, the summation of the vectors must be zero.

In order to calculate, we can use the component method.
  T_{ax}=T_a \cos 270^o=0, T_{ay}=T_a \sin 270^o = -mg  \\ T_{bx}=T_b \cos 0^o=T_b, T_{by}=T_b \sin 0^o=0 \\ T_{cx}=T_c \cos 150^o, T_{cy}=T_c \sin 150^o  
Thus,
T_{ax}+T_{bx}+T_{cx}=T_b+T_c\cos 150^o = 0 \\    T_{ay}+T_{by}+T_{cy}=-mg+T_c\sin 150^o = 0
From the second equation, we have
T_c = \frac{mg}{\sin 150^o}=39.2 \mathrm{N}
Then, plug it in the first equation.
T_b = T_c\sin 150^o = 33.9 \mathrm{N}












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Friday, January 29, 2016

Find the extrema: an optimization problem with Lagrange multipliers

Question:
By using Lagrange multipliers, find the extrema of x^2+2y^2+3z^2 when 3x+2y+z=-1.

Answer:
Using a Lagrange multiplier, we define the following function:
F(x,y,z) =  x^2+2y^2+3z^2 - \lambda (3x+2y+z+1) 
Take each partial derivative.
F_x = 2x-3\lambda = 0  \\ F_y = 4y-2\lambda = 0    \\ F_z = 6z-\lambda = 0   \\ F_{\lambda} = 3x+2y+z+1 = 0  
They have four unknowns and four equations. Solve for each variable.
x=-9/34  \\   y=-3/34  \\   z=1/34  \\   \lambda = -6/34
If you plug x, y, and z, we obtain F = 3/34 as the extremum.

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Finding the extrema; an optimization problem

Question:
Find the extrema of x^2+2y^2+3z^2 when 3x+2y+z=-1.

Answer:
From the condition, we can solve for z.
z = -(3x+2y+1)
Thus, we can define a function.
F(x,y) = x^2+2y^2+3(-(3x+2y+1))
It is supposed to find the extrema of this function. Take the partial derivatives.
\frac{\partial F(x,y)}{\partial x} = 56x+36y+18,  \\   \frac{\partial F(x,y)}{\partial y} = 36x+28y+12,   \\    \frac{\partial^2 F(x,y)}{\partial x^2} = 56,  \\     \frac{\partial^2 F(x,y)}{\partial y^2} = 28,  \\       \frac{\partial^2 F(x,y)}{\partial x \partial y} = 36  
The value of the extremum is found when \frac{\partial F(x,y)}{\partial x} = 0 and \frac{\partial F(x,y)}{\partial y} = 0. Namely,
x = -\frac{9}{34}  \\    y = -\frac{3}{34}  
In order to find if this is maximum or minimum, calculate D = F_{xx}F_{yy}-F^2_{xy}. Then, use the following condition:
If D>0 and F_{xx}>0, then it is a minimum.
If D>0 and F_{xx}<0, then it is a maximum.
If D<0, then it is not a extremum.
If D=0, then it cannot be determined in this method.
For this problem, D = 272>0 and F_{xx}=56>0, so F(-9/34,-3/34) = 3/34 is the minimum.

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Lagrangian function of a sphere on a cylindrical surface

Question:
There is an open cylinder of radius R., which is stationary. A hollow sphere of radius \rho and mass m rolls on the surface without slipping. Find the Lagrangian function.

Answer:
The Lagrangian function is defined as
L = KE - PE
where KE and PE represent kinetic energy and potential energy, respectively.
The mechanical potential energy is the gravitational force \times height, which is taken from the center. The height varies with the angle \theta. Thus,
PE = -mg(R-\rho)\cos\theta
The negative sign indicates the height is directed to the center.
Now the kinetic energy has two parts, translational and rotational kinetic energies.
KE = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 
The translational velocity, v, is along with the trace of circle whose radius is R-\rho. Therefore,
v = (R-\rho)\omega
The angular velocity can be expressed by
\omega = \theta' = \frac{d\theta}{dt}
The moment of inertia of hollow sphere is
I = \frac{2}{3}m\rho^2
Plug everything into above.
L = \frac{1}{2}m(R-\rho)^2 \theta '^2 + \frac{1}{2}\frac{2}{3}m\rho^2 \theta '^2 + mg(R-\rho)\cos\theta 
We can rewrite it as
L = \frac{m\theta'^2}{6}(3R^2 -6R\rho + 5\rho^2)+ mg(R-\rho)\cos\theta

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Thursday, January 28, 2016

Residue theorem and integral of a complex-valued function

Question:
Find the value of
\int_C \frac{dz}{z^2+2iz-4}
The contour, C, is a circle that has center z=1 and radius \sqrt{2}. It is directed positively.

Answer:
The integrand can be reduced in two factors.
I = \int_C \frac{dz}{z^2+2iz-4} = \int_C \frac{dz}{(z+(\sqrt{3}+i))(z-(\sqrt{3}-i))}
The value when the denominator becomes zero is the singular point in the contour. z=-(\sqrt{3}+i) and z=\sqrt{3}-i can be the points, but only z=\sqrt{3}-i is the pole inside contour.
Calculate the residue.
\mathrm{Res}(f(z):\sqrt{3}-i) = \lim_{z\rightarrow \sqrt{3}-i}(z-(\sqrt{3}-i)f(z) \\                            = \lim_{z\rightarrow \sqrt{3}-i}(z-(\sqrt{3}-i)\frac{1}{(z+(\sqrt{3}+i))(z-(\sqrt{3}-i))} \\                            = \lim_{z\rightarrow \sqrt{3}-i}\frac{1}{z+(\sqrt{3}+i)}    \\                            = \frac{1}{2\sqrt{3}}
Then, using the residue theorem, we have the value of integral:
I = 2\pi i \times \frac{1}{2\sqrt{3}} = \frac{\sqrt{3}\pi i}{3}

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Mechanical model of the proton (angular momentum etc.)

Question:
The proton's mass and radius are 1.67 \times 10^{-27} kg and 1.00 \times 10^{-15} m. This system is governed by quantum and classical regimes. Namely, the proton is assumed to have a rotational uniform solid spherical body and a half spin of quantum angular momentum. Find the equatorial velocity of the proton.

Answer:
In terms of classical sense, the moment of inertia of proton is
I = \frac{2}{5}mr^2
Therefore, the angular momentum becomes
L = I\omega = \frac{2}{5}mr^2\omega
For the quantum perspective, the spin angular momentum is given as
S = \frac{1}{2}\hbar
Therefore,
 \frac{2}{5}mr^2\omega  = \frac{1}{2}\hbar
Solve for the angular velocity.
\omega = \frac{1}{2}\hbar \frac{5}{2mr^2}
The equatorial velocity is given by
v = r\omega =  \frac{5\hbar}{4mr} \\       = \frac{5}{4}\frac{1.055\times 10^{-34}\mathrm{Js}}{1.673\times 10^{-27}\mathrm{kg}\times 1.00\times 10^{-15}\mathrm{m}}
Then we have
v = 7.88 \times 10^7 \mathrm{m/s}

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Wednesday, January 27, 2016

Sound intensity

Question:
There is a speaker emitting sound with a uniform power of 250 W. At what distance will the intensity be just below the threshold of pain, which is 1.00 W/m^2?

Answer:
The sound intensity is given by
I = \frac{P}{A}
The sound emitted uniformly in space, so it is spherically propagated. The area of sphere is 4\pi r^2. Thus,
I=\frac{P}{4\pi r^2}
Solve for the distance.
r^2 = \frac{P}{4\pi I} \\     r = \sqrt{\frac{P}{4\pi I}}
Plug in the numbers.
r = \sqrt{\frac{250 (\mathrm{W})}{4\pi \times 1.00 (\mathrm{W/m}^2)}}
Therefore, the distance is
r = 4.46 \mathrm{m}

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Simultaneous ordinary differential equation 1

Question:
Solve for the following simultaneous differential equations:
\left\{ \begin{array}{ll}            \frac{dx}{dt}+ay = e^{bt}  & (1)\\            \frac{dy}{dt}-ax = 0  & (2)           \end{array}    \right.  
where a and b are real numbers.

Answer:
Take differentiation of equation (1).
x'' + ay' = be^{bt} 
Use (2).
x'' + a^2x = be^{bt}
The general solution of this is given when right hand side is equal to zero.
x'' + a^2x = 0
Thus,
x(t) = C_1\cos at + C_2\sin at 
To solve for the particular solution, due to the expression of right hand side, we can assume that it is x=Ae^{bt}. Then, substitute it into the above equation:
(Ae^{bt})'' + a^2(Ae^{bt})' = Ab^2e^{bt} + Aa^2e^{bt} = (a^2 + b^2)Ae^{bt} 
This must be equal to be^{bt}, so
A = \frac{b}{a^2 + b^2} 
The particular solution becomes \frac{b}{a^2 + b^2}e^{bt}. Therefore,
x(t) = C_1\cos at + C_2\sin at + \frac{b}{a^2 + b^2}e^{bt}
Plug this solution into (1) to obtain the solution of y(t).
(C_1\cos at + C_2\sin at + \frac{b}{a^2 + b^2}e^{bt})'+ay=e^{bt}  \\    -C_1a\sin at + C_2a\cos at + \frac{b^2}{a^2 + b^2}e^{bt} + ay = e^{bt} \\   ay = C_1a\sin at - C_2a\cos at - \frac{b^2}{a^2 + b^2}e^{bt} + e^{bt}
The total solution of y becomes
 y(t) = C_1\sin at - C_2\cos at + \frac{a}{a^2 + b^2}e^{bt}

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Normalization of a wave function


Question:
The wave function for the one dimensional harmonic oscillator with the potential energy, \frac{1}{2}kx^2, is given as
\Phi_0 = C\exp(-ax^2)
Find the constant C when the wave function is normalized. Note that you can use the normalized
Gaussian distribution: \frac{1}{\sqrt{2\pi}\sigma}\int^{\infty}_{-\infty}\exp(-\frac{x^2}{2\sigma^2})dx = 1.

Answer:
For the normalized wave function, it has to satisfy:
 \int^{\infty}_{-\infty}\Phi_0\Phi^*_0 dx = C^2\int^{\infty}_{-\infty}\exp(-2ax^2) dx = 1
where \Phi^*_0 is the complex conjugate of \Phi_0. Compare this with the Gaussian normal distribution.
2a = \frac{1}{2\sigma^2}  \\    \sigma = \frac{1}{\sqrt{4a}}
Plug it in the formula of Gaussian distribution.
\frac{1}{\sqrt{2\pi}(1/\sqrt{4a})}\int^{\infty}_{-\infty}\exp(-\frac{x^2}{2(1/4a)})dx = 1
We can see
C^2 = \frac{1}{\sqrt{2\pi}(1/\sqrt{4a})} = \frac{1}{\sqrt{2\pi}}\sqrt{4a} \\            = \sqrt{\frac{2a}{\pi}}
Therefore,
C = (2a/\pi)^{1/4}

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Tuesday, January 26, 2016

Integral using Beta and Gamma functions

Question:
Find the value of
\int^{\infty}_{-\infty}\frac{dx}{x^6+1}

Answer:
We can reduce the above integral into
\int^{\infty}_{-\infty}\frac{dx}{x^6+1} = 2\int^{\infty}_{0}\frac{dx}{x^6+1}
Replace x with t^{\frac{1}{6}}. Then, we have dx=\frac{1}{6}t^{-\frac{5}{6}}dt. Thus,
\int^{\infty}_{0}\frac{dx}{x^6+1} = \frac{1}{6}\int^{\infty}_{0}\frac{t^{-\frac{5}{6}}}{t+1}dt
Remember that the Beta function is defined as follows:
B(p,q) = \int^{\infty}_{0}\frac{x^{p-1}}{(1+x)^{p+q}}dx \\ B(p,q)=B(q,p)=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}
Therefore, the above expression becomes
\frac{1}{6}\int^{\infty}_{0}\frac{t^{-\frac{5}{6}}}{t+1}dt = \frac{1}{6}B(\frac{1}{6},\frac{5}{6}) \\    = \frac{1}{6}\frac{\Gamma(\frac{1}{6})\Gamma(\frac{5}{6})}{\Gamma(\frac{1}{6}+\frac{5}{6})} \\    =  \frac{1}{6}\Gamma(\frac{1}{6})\Gamma(\frac{5}{6})  
since \Gamma(1)=1.
Recall the following formula:
\Gamma(s)\Gamma(1-s) = \frac{\pi}{\sin\pi s}  
The parameter satisfies 0<s<1.
Therefore,
\frac{1}{6}\Gamma(\frac{1}{6})\Gamma(\frac{5}{6}) = \frac{1}{6}\frac{\pi}{\sin(\pi/6)} = \frac{\pi}{3}
Hence,
\int^{\infty}_{-\infty}\frac{dx}{x^6+1} = \frac{2\pi}{3}

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Newton's equation of motion

Question:
The objects A and B have masses, 20.0 kg and 10.0 kg, respectively. They are placed on a frictionless surface. The force (29.4 N) is acted as shown and these objects move together.
(1) Find the acceleration of the objects.
(2) What is the force exerting B from A?

Answer:
From the second figure, the forces exerting B from A and vice versa are equal due to the action and re-action pair.

(1) You can set up the Newton's equation for object A:
\sum F_A = F - N = m_A a
The equation for object B is:
\sum F_B = N = m_B a
Notice that the acceleration of each object is the same since they move together. Solve for the acceleration from them. Plug N=m_B a into F - N = m_A a.
F - m_B a = m_A a \\     (m_A + m_B)a = F  \\    a = \frac{F}{m_A + m_B}
Therefore,
a = \frac{29.4 \mathrm{N}}{20.0 \mathrm{kg} + 10.0 \mathrm{kg}} = 0.98 \mathrm{m/s^2}

(2) From the solution in (1), we can find the force exerting B from A, which is N in the figure. Thus,
N = m_B a = 10.0\mathrm{kg}\times 0.98\mathrm{m/s^2} = 9.8\mathrm{N}

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Monday, January 25, 2016

Differentiation with trigonometric and logarithmic functions

Question:
Here is a function of x.
y=\alpha \cos x + 2 - \cos x \log\frac{1+\cos x}{1-\cos x} 
\alpha is a constant, and the range of x is 0<x<\pi.
Calculate \frac{d^2 y}{dx^2}+\frac{\cos x}{\sin x}\frac{dy}{dx}.

Answer:
Define each term.
y_1 =  \alpha \cos x \\     y_2 = 2 \\     y_3 = \log\frac{1+\cos x}{1-\cos x}
Calculate each derivative.
y_1' = -\alpha \sin x  \\   y_2' = 0 \\  
The above derivatives are easy, but the third one is a little complicated:
 y_3' = (\log|1+\cos x| - \log|1-\cos x|)' \\          = \frac{(1+\cos x)'}{1+\cos x} - \frac{(1-\cos x)'}{1-\cos x}   \\          = \frac{-\sin x}{1+\cos x} - \frac{\sin x}{1-\cos x}   \\         = \frac{-\sin x(1-\cos x) - \sin x(1+\cos x)}{1 - \cos^2x}  \\         = \frac{-2\sin x}{\sin^2 x}   \\         = \frac{-2}{\sin x}
Since y=y_1+y_2-\cos x y_3, the derivative becomes y'=y_1'+y_2'-[(\cos x)' y_3+\cos x y_3']. Namely,
\frac{dy}{dx} = -\alpha\sin x + \sin x \log\frac{1+\cos x}{1-\cos x} + \frac{2\cos x}{\sin x}
The second derivative is calculated as follows:
\frac{d^2y}{dx^2} = -\alpha\cos x + \cos x \log\frac{1+\cos x}{1-\cos x} + \sin x\frac{-2}{\sin x} + \frac{2((\cos x)'\sin x - \cos x(\sin x)')}{\sin^2x}  \\            = -\alpha\cos x + \cos x \log\frac{1+\cos x}{1-\cos x} - 2 - \frac{2}{\sin^2x}
You can notice that the first three terms of above is equal to -y. Therefore,
 \frac{d^2y}{dx^2} = -y -\frac{2}{\sin^2 x} 
The next term becomes
\frac{\cos x}{\sin x}\frac{dy}{dx} = -\alpha\cos x + \cos x\log\frac{1+\cos x}{1-\cos x} + \frac{2\cos^2 x}{\sin^2 x} 
This can be rewrite as
\frac{\cos x}{\sin x}\frac{dy}{dx} = -y + 2 + \frac{2\cos^2 x}{\sin^2 x} 
Thus, we have
\frac{d^2 y}{dx^2}+\frac{\cos x}{\sin x}\frac{dy}{dx} = -2y

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Sunday, January 24, 2016

A problem with trigonometric addition theorem

Question:
Show that \frac{\sin(\alpha + \beta)\sin(\alpha - \beta)}{\sin\alpha - \sin\beta}=\sin\alpha + \sin\beta.

Answer:
Use \sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \sin\beta\cos\alpha.
The given expression will become
\frac{\sin(\alpha + \beta)\sin(\alpha - \beta)}{\sin\alpha - \sin\beta} = \frac{(\sin\alpha\cos\beta + \sin\beta\cos\alpha)(\sin\alpha\cos\beta - \sin\beta\cos\alpha)}{\sin\alpha - \sin\beta}      = \frac{\sin^2\alpha\cos^2\beta - \sin^2\beta\cos^2\alpha}{\sin\alpha - \sin\beta}

Use \sin^2\theta+\cos^2\theta=1 to eliminate \cos^2\alpha and \cos^2\beta.
\frac{\sin^2\alpha\cos^2\beta - \sin^2\beta\cos^2\alpha}{\sin\alpha - \sin\beta}        = \frac{\sin^2\alpha(1-\sin^2\beta) - \sin^2\beta(1-\sin^2\alpha)}{\sin\alpha - \sin\beta}       =  \frac{\sin^2\alpha-\sin^2\alpha\sin^2\beta - \sin^2\beta + \sin^2\beta\sin^2\alpha}{\sin\alpha - \sin\beta}      = \frac{\sin^2\alpha - \sin^2\beta}{\sin\alpha - \sin\beta}      = \frac{(\sin\alpha + \sin\beta)(\sin\alpha - \sin\beta)}{\sin\alpha - \sin\beta}      = \sin\alpha + \sin\beta
QED

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Trigonometric functions with a parametric representation

Question:
Here is a set of simultaneous equations, \sin x = \cos y = \sin (y-x). The ranges are 0 < x < \frac{\pi}{2} and 0 < y < \frac{\pi}{2}. Find x and y.

Answer:
It is easier to equate the equations with a parameter. Thus,
\sin x = \cos y = \sin (y-x) = t
From the conditions, we can notice that \sin x = \cos y = t > 0.  If we use \sin^2\theta + \cos^2\theta = 1, we can have the following expressions:
\cos x = \sqrt{1-t^2};  \sin y = \sqrt{1-t^2}
Now, we can rewrite \sin (y-x) as follows:
\sin (y-x) = \sin y\cos x - \sin x\cos y  \\                     = \sqrt{1-t^2}\cdot\sqrt{1-t^2}-t\cdot t  \\                    = (1-t^2) - t^2  \\                    = 1 - 2t^2
Since \sin(y-x) = t, we have
1 - 2t^2 = t
Therefore,
2t^2 + t - 1 = 0  \\     (2t - 1)(t - 1) = 0
The solution is that t=\frac{1}{2} because t>0.
This gives \sin x = \cos y = \frac{1}{2}. Hence we obtain x = \frac{\pi}{6} and y = \frac{\pi}{3}.

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Saturday, January 23, 2016

Trigonometric functions 2

Question:
When \tan\theta = -2, find \cos 2\theta and \sin 2\theta.

Answer:
This might be a little tricky, but if you come up with a relationship between \tan^2 \theta and \cos^2 \theta first. Then, \cos^2 \theta can be converted into \cos 2\theta.

Remember
1+\tan^2\theta = \frac{1}{\cos^2\theta} 
This can be derived from \sin^2\theta + \cos^2\theta = 1 and \tan\theta = \frac{\sin\theta}{\cos\theta}.

Since \tan\theta = -2, we have
\cos^2\theta = \frac{1}{1+\tan^2\theta} = \frac{1}{1+(-2)^2} = \frac{1}{5} 

Now, recall the double-angle formula as for cosine:
\cos 2\theta = 2\cos^2\theta -1
Plug above result into this.
\cos 2\theta = 2\times\frac{1}{5} -1 = -\frac{3}{5}
Also remember
\sin 2\theta = 2\sin\theta\cos\theta
Modify the right hand side.
\sin 2\theta = 2\sin\theta\cos\theta \times \frac{\cos\theta}{\cos\theta} = 2\tan\theta\cos^2\theta
Thus, we have
 \sin 2\theta = 2\times(-2)\times\frac{1}{5}=-\frac{4}{5}

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Let's start with tangent!

Question:
Calculate (\tan 15^o + \sqrt{3})^2 without a calculator.

Answer:
It is not easy to obtain \tan 15^o without a calculator. In this case, you may want to come up with angles which can easily convert into specific numbers, such as 30, 45, 60, 90, etc. In fact, 15 = 45 - 30, so \tan 15^o = \tan(45^o - 30^o). Thus, we can use the addition theorem.
\tan(45^o - 30^o) = \frac{\tan 45^o - \tan 30^o}{1 + \tan 45^o \tan 30^o}
If we use typical triangles, we know \tan 45^o = 1 and \tan 30^o = 1/\sqrt{3}. Therefore,
\tan 15^o = \frac{1 - 1/\sqrt{3}}{1 + 1 \times 1/\sqrt{3}}
Multiply \sqrt{3} by both numerator and denominator.
= \frac{1 - 1/\sqrt{3}}{1 + 1/\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}
= \frac{\sqrt{3}-1}{\sqrt{3}+1}
= \frac{\sqrt{3}-1}{\sqrt{3}+1} \times \frac{\sqrt{3}-1}{\sqrt{3}-1}
= \frac{4 - 2\sqrt{3}}{2} = 2 - \sqrt{3}
Hence, we have
(\tan 15^o + \sqrt{3})^2 = 2^2 = 4

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