Problems

If \(\phi\) is the angle between the radius vector and the tangent to the curve \(f(r,\theta)=0\), prove that \(\tan\phi=r\frac{d\theta}{dr}\). Prove that, if the tangents at \(P, Q\), two points on the curve \(r=a(1-\cos\theta)\), are parallel, the chord \(PQ\) subtends an angle \(2\pi/3\) at the pole.

Prove that if \(\rho\) is the radius of curvature at any point of a curve \[ \frac{1}{\rho^2} = \left(\frac{d^2x}{ds^2}\right)^2 + \left(\frac{d^2y}{ds^2}\right)^2. \] Prove also that \[ \frac{1}{\rho^4}\left\{1+\left(\frac{d\rho}{ds}\right)^2\right\} = \left(\frac{d^3x}{ds^3}\right)^2 + \left(\frac{d^3y}{ds^3}\right)^2. \]

Integrate

1. [(i)] \(\int_0^1 \frac{dx}{x^2+x+1}\);
2. [(ii)] \(\int_0^\pi x^2\sin x dx\);
3. [(iii)] \(\int_0^\pi \frac{dx}{3+2\cos x}\) when \(r=1,2\).

Find a formula of reduction for \[ \int \frac{x^m dx}{(a^2+x^2)^n}. \]

Trace the curve \(x^2(x^2-a^2)+y^2(x^2+a^2)=0\), and find the area of a loop.

Prove that \[ \tan 20^\circ \tan 30^\circ \tan 40^\circ = \tan 10^\circ. \] If \(A+B+C=90^\circ\), prove that \[ \frac{\tan A}{\tan C} = \frac{1-\cos 2A+\cos 2B+\cos 2C}{1+\cos 2A+\cos 2B-\cos 2C}. \]

The lines joining the angular points of a triangle \(ABC\) to the middle points of the opposite sides intersect in a point \(O\); prove that \[ OA^2+OB^2+OC^2 = \frac{8}{9}R^2(1+\cos A \cos B \cos C), \] where \(R\) is the radius of the circumcircle.

Prove that the sum of \(n-1\) terms of the series \[ \tan\theta\tan 2\theta + \tan 2\theta\tan 3\theta + \tan 3\theta\tan 4\theta + \dots \] is equal to \(\tan n\theta\cot\theta-n\). Sum the series \[ \cos\alpha + \frac{1}{3!}\cos(\alpha+2\beta) + \frac{1}{5!}\cos(\alpha+4\beta) + \dots \text{ to infinity}. \]

Resolve \(x^{2n}-2x^ny^n\cos n\theta+y^{2n}\) into factors. Prove that \[ \sin n\phi = 2^{n-1}\sin\phi \sin\left(\phi+\frac{\pi}{n}\right)\dots\sin\left(\phi+\frac{n-1}{n}\pi\right). \]

Prove that the equation of the straight line joining the feet of the perpendiculars from the point \((h,k)\) on the lines \(ax^2+2bxy+cy^2=0\) is \[ x(ah-ch+2bk)+y(ck-ak+2bh)+ak^2-2bhk+ch^2=0. \]

Find the equation of the normal at \(P\) to the parabola \(y^2=4ax\) in the form \[ y = mx - 2am - am^3, \] and, if the normal cuts the parabola again at \(Q\), prove that \(PQ=4a(1+m^2)^{3/2}/m^2\).