Mathematical Methods for Physics and Engineering : A Comprehensive Guide

1.8 EXERCISES

(a) the sum of the sines ofπ/3andπ/6,

(b) the sine of the sum ofπ/3andπ/4.

1.8 The following exercises are based on the half-angle formulae.

(a) Use the fact that sin(π/6) = 1/2 to prove that tan(π/12) = 2−

√

3.

(b) Use the result of (a) to show further that tan(π/24) =q(2−q)where

q^2 =2+

√

3.

1.9 Find the real solutions of

(a) 3 sinθ−4cosθ=2,

(b) 4 sinθ+3cosθ=6,

(c) 12 sinθ−5cosθ=− 6.

1.10 Ifs= sin(π/8), prove that

8 s^4 − 8 s^2 +1=0,

and hence show thats=[(2−

√

2)/4]^1 /^2.

1.11 Find all the solutions of

sinθ+sin4θ=sin2θ+sin3θ

that lie in the range−π<θ≤π. What is the multiplicity of the solutionθ=0?

Coordinate geometry

1.12 Obtain in the form (1.38) the equations that describe the following:

(a) a circle of radius 5 with its centre at (1,−1);

(b) the line 2x+3y+ 4 = 0 and the line orthogonal to it which passes through

(1,1);

(c) an ellipse of eccentricity 0.6 with centre (1,1) and its major axis of length 10

parallel to they-axis.

1.13 Determine the forms of the conic sections described by the following equations:

(a) x^2 +y^2 +6x+8y=0;

(b) 9x^2 − 4 y^2 − 54 x− 16 y+29=0;

(c) 2x^2 +2y^2 +5xy− 4 x+y−6=0;

(d)x^2 +y^2 +2xy− 8 x+8y=0.

1.14 For the ellipse

x^2

a^2

+

y^2

b^2

=1

with eccentricitye, the two points (−ae,0) and (ae,0) are known as its foci. Show

that the sum of the distances fromanypoint on the ellipse to the foci is 2a.(The

constancy of the sum of the distances from two fixed points can be used as an

alternative defining property of an ellipse.)

Partial fractions

1.15 Resolve the following into partial fractions using the three methods given in
section 1.4, verifying that the same decomposition is obtained by each method:

(a)

2 x+1

x^2 +3x− 10

, (b)

4

x^2 − 3 x

.