Maths for Physics 1-Trigonometric Functions, Logarithms and Exponentials PDF

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Mathematics for Physics I

Workbook 3

Trigonometric Functions, Logarithms and Exponentials The most common trigonometric functions are probably sine and cosine, defined with the aid of the unit circle (x = R cos θ, y = R sin θ, R = 1). They are useful ratios in right-angled triangle and the angles are measured either in degrees or in radians. It is not necessary to learn all the various trigonometric formulae, but you should know what is available. In particular, when integrating trigonometric functions, knowing your options about how to rewrite the function, often solves the integral for you.

Figure 1: Polar representation of Sine and Cosine, x = cos θ and y = sin θ. The double angle formulae, as well as the basic formulae for sin (α ± β), cos (α ± β), and tan (α ± β) are worth learning. Also, the Pythagoras-type identities and a few special  √  √  values that can be deduced from right angled triangles with sides 1, 1, 2 or 1, 3, 2 . This week we will also be working with logarithmic and exponential functions.

Figure 2: y = ex and y = ln x, symmetric with respect to the line y = x.

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Mathematics for Physics I One of the most common integrals in physical examples is a simple function Z x ds , x > 0. s 1 This function comes up in so many problems, that people have made tables of it and given it a name: it’s called natural logarithm, ln x.

Figure 3: ln x = and the x-axis.

Rx 1

dx′ , x′

is the area bounded by the curve y =

1 , x′

the lines x′ = 1, x′ = x,

This week we build on the material presented at lectures in week 1 and 21 : • Sine, Cosine and Tangent • Polar representation of Sine and Cosine (p24-27) • Trigonometric identities (p28-32) • Trigonometric equations • Inverse trigonometric functions • Algebra with Exponentials and Logarithms (p7-9) • Exponentials and Logarithms in physics (p11-14) • Odd and Even Functions

1

The page numbers in brackets refer to the text book Foundation Mathematics for the Physical Sciences by K F Riley & M P Hobson (Cambridge university Press, 2011).

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Mathematics for Physics I

Trigs and Logs: Problems 3

Sine and Cosine: some exact values and basic relations Ex3.1 Use the polar representation (R = 1), i.e go round the unit circle, to find the following angles, and evaluate     c) cos 3π ; d) sin − π2 . b) cos − π2 ; a) sin 2π ; 2 π Answers: a) sin π2 = 1; b) cos − π2 = 0; c) cos 3π 2 =0; d) sin − 2 = −1.









Ex3.2 Use the polar representation (R = 1), i.e go round the unit circle, to find the following angles, and evaluate a) cos π2 ;

b) sin − 2π ;

c) cos 2π;

d) sin 2π;

e) sin 5π;

; f ) cos 3π 2

g) sin 5π ; 2

h) cos 25π.

Answer: a) 0; b) −1; c) 1; d) 0; e) 0; f ) 0; g) 1; h) −1. Ex3.3 Find the following angles in xy-plane, and evaluate

Answers: a) cos π4 =  d) cos − 5π 4

; c) sin 7π 4

b) sin π4 ;

a) cos 4π ;

√ 2 ; 2

  . d) cos − 5π 4

√ √  π 2 b) sin π4 = √22 ; c) sin 7π = − = sin − 2 ; 4 4 2 = cos 3π 4 =− 2 .

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Mathematics for Physics I Ex3.4 Find the following angles, and evaluate     a) cos 4π ; ; c) cos − 7π b) sin − 4π ; 4 Answers: a)

3.1

1 √ ; 2

b) − √12 ; c)

√1 ; 2

d) − √12 .

Find the coordinates for the point on the unit circle that is defined by the angle µ = π3 from positive x-axis.

Ex3.5 Find the following angles, and evaluate   ; c) cos 7π b) cos − 3π ; a) sin π3 ; 3 ; e) sin 13π 3

Answers: a) 3.2

. d) cos 5π 4

√ 3 ; 2

  f ) sin − 2π ; 3

b) 21 ; c) 12 ; d) − 21 ; e)

  d) cos − 4π ; 3

g) sin 2π ; 3 √ 3 ; 2

f) −

√ 3 ; 2

h) cos 2π . 3

g)

√ 3; 2

h) − 21 .

Find the coordinates for the point on the unit circle that is defined by the angle φ = π6 from positive x-axis.

Ex3.6 Find the following angles, and evaluate a) cos π6 ;

b) sin 6π ;

; c) cos 17π 6

; e) sin 11π 6

  f ) sin − 11π ; 6

  g) cos − 11π 6

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  d) cos − 17π ; 6

  h) sin − 17π . 6

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Mathematics for Physics I √ 3 ; 2

Answers: a)

b) 21 ; c) −

√

3 ; 2

d) −

√ 3 ; 2

e) − 21 ; f ) 12 ; g)

√ 3 ; 2

h) − 12 .

cos (−φ) = cos φ. sin (−φ) = − sin φ.

3.3

Verify the relations above for an angle v by means of a sketch in xy-plane. cos sin

π

 − φ = sin φ.

2

 − φ = cos φ.

2

π

3.4

Verify the relations above for an angle u by means of a sketch in xy-plane.     sin β = 31. Ex3.7 Evaluate cos π2 − β and cos β − π2 if    π 1 π Answer: cos 2 − β = 3 and cos β − 2 = cos π2 − β = 13 .     3.5 Evaluate sin 2π − β and sin β − π2 if cos β = 41 . cos (φ + π) = − cos φ. sin (φ + π) = − sin φ.

3.6

Show that cos (π − α) = − cos α, and sin (π − α) = sin α.

Ex3.8 Evaluate the following ; a) cos 5π 6 Answers: a) − 3.7

√ 3 ; 2

b) sin 5π ; 6 b) 21 ; c) − √12 ; d)

c) cos 3π ; 4

. d) sin 3π 4

√1 . 2

Show that cos2 θ + sin2 θ = 1, for any angle θ. Hint: Graph and Pythagoras.

Ex3.9 Find sin φ if a) cos φ = 13 ;

b) cos φ = 71. √

√

Answers: a) sin2 φ = 1 − cos2 φ which gives sin φ = ± 2 32 ; b) ± 4 7 3 . Ex3.10 Sketch in the same graph the following curves   a) y = sin x, and y = sin x + 6π ; b) y = sin x, and y = sin 2x;   c) y = sin x, and y = sin 2x + 6π .

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Mathematics for Physics I Answers:

3.8

Sketch in the same graph the following curves   e) y = cos x, and y = cos 2x; a) y = cos x and y = cos x + 3π ; c) y = cos x, and y =

1 2

  cos x − π3 .

Tangent and Cotangent Tangent (and cotangent) are somewhat more complicated trigonometric functions. The first thing to do when working with tangent or cotangent is to express them in terms of the much simpler trigonometric functions sine or cosine, and take it from there.

3.9

sin x tan x =cos x,

x 6=

x cot x = cos sin x ,

x 6= nπ, where n is an integer.

π 2

+ nπ;

Evaluate the following a) tan π4 ;

c) tan 13π ; 4

b) cot π3 ;

  d) cot − π6 .

tan (x + nπ) = tan x; cot (x + nπ) = cot x, where n is an integer.

3.10*

Sketch tan φ for −2π ≤ φ ≤ 2π . Hint: Make use of the definition of tan x and mark on the sketch some special values, ie the vertical asymptotic lines when cos x = 0, the zeros of sin x, etc.

3.11*

Sketch in the same graph the curves y = tan 2x and y = tan x, where −π ≤ x ≤ 3π .

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Mathematics for Physics I

Trigonometric identities cos (a + b) = cos a cos b − sin a sin b sin (a + b) = sin a cos b + cos a sin b

3.12

Derive the corresponding identities for cos (a − b) and sin (a − b). Hint: substitute b by −b in the formulae above and use the properties sine and cosine of negative angle.

3.13

Starting from the identity sin (α + β) = sin α cos β + cos α sin β, and making use of the basic properties of the trigonometric functions (but not the identities in the box above) prove the following a) sin (α − β) = sin α cos β − cos α sin β; b) cos (γ + δ) = cos γ cos δ − sin γ sin δ; tan φ+tan θ c)∗ tan (φ + θ) = 1−tan ; φ tan θ

d)∗ sin µ + sin ν = 2 sin 21 (µ + ν) cos 21 (µ − ν). Ex3.11 If sin u = 54 , cos u = − 53 , and sin v = 53 , cos v = 45 , evaluate sin (u + v). 7 Answer: 25 . 3.14

By considering

π 12

=

π 3

− 4π, find the exact values of

π ; a) sin 12

π. b) cos 12

3.15

By considering the formulae for cos (u + v) and sin (u + v), and putting u = v , derive the double angle formulae cos 2u = cos2 u − sin2 u and sin 2u = 2 cos u sin u respectively.

3.16

Show that for all φ the following relations are true a) cos 2φ = 1 − 2 sin2 φ;

b) cos 2φ = 2 cos2 φ − 1;

c) cos 3φ = 4 cos3 φ − 3 cos φ;

d) sin 4φ = 4 sin φ cos φ − 8 sin3 φ cos φ;

e) tan 2φ = 3.17

2 tan φ , 1−tan2 φ

φ 6=

π 4

+ nπ,

n = 0, 1, 2, 3...

From the formulae sin2 α + cos2 α = 1, and cos2 α − sin2 α = cos 2α, derive the following b) sin2 φ =

a) cos2 φ = 21 (1 + cos 2φ) ; School of Physics and Astronomy

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1 2

(1 − cos 2φ) . September 19, 2014

Mathematics for Physics I 3.18

By making use of the half-angle formulae, or otherwise, evaluate the exact value π . of cos 8π , sin π8 , and cos 16 You should also recognize the related functions:

sec x =

1 cos x

cosec x = csc x =

1 sin x

Some basic trigonometric equations cos x = cos φ is equal to x = φ + 2πn or x = −φ + 2πn, where n is an integer.

Rather than remembering these general solutions, often the most efficient method for solving trigonometric equations is by means of a graph, as shown in the example below. Ex3.12 Find all φ that satisfy the equation cos x = 21. Answer: First draw a sketch in xy-plane of the unit circle and the line y = 21.

Mark the points of intersection P1 and P2 between the line y = 21 and the perimeter of the unit circle. Then the (principal) solution to the equation is given by the angles between the positive x-axis and the lines y = OPi . Thus, φ = ±3π is the principal solution, and all the solutions are given by φ = ±3π + 2πn.

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Mathematics for Physics I Ex3.13 Solve the following equations. Sketch each case. a) cos x = cos π6 ; d) cos x =

√1 ; 2

b) cos x = cos 4;

c) cos x = 0;

e) cos x = − 12 ;

f ) cos x = −

√ 3 . 2

Answers: a) x = ± π6 +2nπ; b) x = ±4+2nπ; c) x = ± 2π +2nπ; d) x = ±4π+2nπ; + 2nπ; f ) x = ± 5π e) x = ± 2π + 2nπ. 3 6 Similarly we have for the sine function:

sin x = sin φ is equal to x = φ + 2πn or x = π − φ + 2πn, where n is an integer.

3.19

Sketch the above relations for some angle φ to verify and understand the meaning of these formulae.

Ex3.14 Solve the following equations. Sketch each case. √ 3 ; 2

a) sin x = sin 5π ;

b) sin x =

d) sin x = − 21 ;

e) sin x = 21 ;

c) sin x = 0; f ) sin x = sin 2.

+ 2nπ; b) x = 3π + 2nπ and x = 2π + 2nπ; Answers: a) x = π5 + 2nπ and x = 4π 5 3 7π + 2nπ; c) x = nπ; d) x = 6 + 2nπ and x = 11π 6 + 2 nπ; f ) x = 2+ 2nπ and x = π − 2+ 2nπ. e) x = π6 + 2nπ and x = 5π 6 Ex3.15 Solve the following equations    a) sin 6x = 12 ; b) cos 2 x + π6 = π + nπ and x = Answers: a) x = 36 3  π 9π 17π ; 20 ; 20 . c) x = 20

5π 36

√1 ; 2

+

nπ ; 3

  c) cos 5x − 4π = 1, where 0 ≤ x ≤ π. b) x = ± π8 −

π 6

+ nπ;

Ex3.16 Solve the following equations a) cos2 φ = 12 ; d) sin 2φ =

√

2 cos φ;

b) sin2 φ = 43;

c) 2 cos2 φ − 3 cos φ + 1 = 0;

e) sin 2φ = 2 sin φ;

f ) cos 2φ = cos φ.

Answers: a) cos φ = ± √1 2 , and φ = ± 4π +n π2 . Alternatively, use cos 2φ = 2 cos2 φ − 1 to rewrite the equation as cos 2φ = 0 and solve it; b) φ = ± π3 + n 2π; c) factorize or solve the quadratic with x = cos φ, then cos φ = 1 or cos φ = 12 , and φ = n2π or φ = ± π3 + 2nπ;  √  d) use sin 2φ = 2 sin φ cos√φ to get cos φ 2 sin φ − 2 = 0, thus cos φ = 0 or 2 sin φ = 2, solved by φ = 2nπ and φ = π4 + 2nπ, 3π 4 + 2nπ; + 2πn; e) cos φ = 0 and cos φ = − 12 giving φ = ± π2 + πn and φ = ± 2π 3 f ) use cos 2φ = 2 cos2 φ − 1 to get 2 cos2 φ − cos φ − 1 = 0, solve it as a quadratics with cos φ as the variable. This gives cos φ = 1 or −21, and thus φ = ±2nπ and φ = ± 2π 3 + 2πn.

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Mathematics for Physics I Ex3.17 Solve the following equations b) sin2 θ + cos θ = 45 .

a) 2 cos2 θ − sin θ = 1; Answers: a) θ =

π 6

+ 2nπ, or θ =

5π + 2nπ, 6

or θ =

3π 2

+ 2nπ; b) θ = ±3π + 2nπ.

Ex3.18 Solve the following equations a) cos 2α + cos α + 1 = 0;

b) cos γ sin γ = 0;

c) 2 cos 2ν + 4 sin ν = 3;

d) cos 2µ = cos2 µ + 3 sin µ.

+ 2nπ, or α = 4π + 2nπ; b) γ = Answers: a) α = π2 + nπ, or α = 2π 3 3 π 5π or γ = nπ; c) ν = 6 + 2nπ or ν = 6 + 2nπ ; d) µ = nπ . 3.20

π 2

+ nπ

Find all real solutions to the following equations  √    a) sin x + π6 + sin x − π6 = 23; b) cos x + cos 2x + cos 3x = 0.

Ex3.19 Solve graphically the following equations √ a) sin x + cos x = 0; b) sin x − 3 cos x = 0; Answers: a) x =

3π 4

+ nπ; b) x =

π 3

+ nπ; c) x =

c) 5π 6

√

3 sin x + cos x = 0.

+ nπ.

The Equation a sin x + b cos x = c To solve equations of this type, you may want to use the method of auxiliary angle. The main idea of this method is to find an angle ϕ such that you can rewrite the expression a sin x + b cos x on the form A sin (x + ϕ), where A is a constant. That transforms the problem into solving the much simpler equation sin (x + ϕ) = k, k = c/A. Use the formula for addition of angles, to express sin (x + ϕ) = sin x cos ϕ + cos x sin ϕ. Define cos ϕ = a and sin ϕ = b, and we have sin (x + ϕ) = a sin x + b cos x. The condition for such and angle ϕ to exist is that a2 + b2 = 1, because sin2 x + cos2 x = 1 for any angle. Therefore, to √ be successful in rewriting our equation, we need to break out the factor a2 + b2 . The equation a sin x + b cos x = c is equivalent with   √ a b 2 2 sin x + √ cos x = c, a +b √ a 2 + b2 a 2 + b2 which gives √

a b c . sin x + √ cos x = √ a 2 + b2 a 2 + b2 a 2 + b2

Now we can determine the auxiliary angle ϕ such that cos ϕ = September 19, 2014

√ a a2 +b2

and 10

sin ϕ = √a2b+b2 . School of Physics and Astronomy

Mathematics for Physics I The given equation can then be written on the form sin (x + ϕ) = √

a2

c . + b2

This is a very general technique that will come handy in many applications in physics, especially in the context of waves and wave propagation of any kind. The functions of the type y = C1 sin ωx + C2 cos ωx, where C1 , C2 and ω are real constants, can be rewritten on the form y = A sin (ωx + φ), where the constants are the amplitude A and phase angle φ.

3.21

Show that the expression a sin x+b cos x can be expressed as A sin (x + φ), where A is a constant and φ is some angle.

Ex3.20 Rewrite the function y = 5 sin x + 5 cos x as y = A sin (ωx + φ), where A, ω, and φ are real constants. Hence solve the equation 5 sin x + 5 cos x = 0. Answer: the phase angle φ must satisfy cos φ = sin φ = √12 , which gives φ = π4 ,  √   √  and y = 5 2 sin x + π4 where A = 5 2, and ω = 1; sin x + 4π = 0 when x + π4 = ±nπ which gives x = − 4π ± nπ.

Ex3.21 Solve the equation cos 2x − sin 2x = 1. Answer: This can be solved by expressing it on for instance the form   3π 1 sin 2x + = √ 4 2 and solving it for x. Or simply realizing that the only possible solutions to this equations must be that 

cos 2x = 1 sin 2x = 0

or



cos 2x = 0 sin 2x = −1

Either way the solutions are x = 3π4 + nπ or x = nπ. Can you think of yet another way of solving this equation? 3.22

Rewrite the following functions as a sin-function, and determine the amplitude and the phase angle in each case a) y =

√

3 sin 3x + cos 3x;

c) y = sin x −

√

b) y = sin 2x + cos 2x; d) y = − 12 sin 5x −

3 cos x;

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√ 3 2

cos 5x.

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Mathematics for Physics I

Inverse Trigonometric Functions y = arcsin (x)

x = sin (y),

−1 ≤ x ≤ 1

− π2 ≤ y ≤ 2π.

y = arccos (x)

x = cos (y),

−1 ≤ x ≤ 1

0 ≤ y ≤ π.

y = arctan (x)

x = tan (y),

−∞ ≤ x ≤ ∞

y = arccot(x)

x = cot (y),

−∞ ≤ x ≤ ∞

−2π ≤ y ≤ π2 . 0 ≤ y ≤ π.

3.23

Sketch the graphs y = arcsin (x) and y = arccos (x).

3.24

Sketch the graphs y = arccot(x) and y = arctan (x).

Ex3.22 Write sin θ = 43 in inverse-relation notation. Answer: θ = arcsin 34 . 3.25

Write the following in inverse-relation notation a) cos φ = −1;

c) cot ζ = 21 .

b) tan α = −2;

Ex3.23 Find the principal value of each of the following a) arcsin 0;

b) arccos (−1);

Answers: a) 0; b) π; c) tan x = 3.26

sin x cos x

=

√ c) arctan 3;

√ d) arccot 3.

√ 3 which gives x = π3 ; d) 6π .

Find the principal value of each of the following a) arcsin (−1);

b) arccos 0;

c) arctan (−1);

Ex3.24 Find the general value of each of the following  √  √ a) arcsin 23; b) arccos − 22 ; c) arctan √13 ; Answers: a) c)

3.27

π 2π 3π 3 + 2nπ and 3 + 2nπ; b) 4 + 2nπ π 7π + 2nπ; d) π4 + nπ. 6 + 2nπ and 6

and

5π 4

d) arccot0.

d) arccot1.

+ 2nπ;

Find the general value of each of the following      √  a) arcsin −21 ; b) arccos − 12 ; c) arctan − 3 ;

d) arccot0.

Ex3.25 Evaluate each of the following 

a) sin arcsin

1 2



 √        ; b) cos arccos −21 ; c) sin arcsin −21 ; d) cos arccos 23 .

Answers: a) 21 ; b) − 12 ; c) − 12 ; d) September 19, 2014

√ 3 2 .

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Mathematics for Physics I 3.28

Evaluate each of the following h  √ i h  √ i a) tan [arctan (−1)]; b) cos arcsin − 2 2 ; c) sin arccos − 23 ; d) tan (arcsin 0).

Ex3.26 Evaluate each of the following           . ; d) arccos tan − 5π a) arcsin sin 3π ; b) arc...