Articles

3.E: Identities (Exercises) - Mathematics


These are homework exercises to accompany Corral's "Elementary Trigonometry" Textmap. This is a text on elementary trigonometry, designed for students who have completed courses in high-school algebra and geometry. Though designed for college students, it could also be used in high schools. The traditional topics are covered, but a more geometrical approach is taken than usual. Also, some numerical methods (e.g. the secant method for solving trigonometric equations) are discussed.

3.1 Exercises

3.1.1 We showed that (;sin; heta ~=~ pm,sqrt{1 ~-~ cos^2 ; heta}; ) for all ( heta ). Give an example of an angle ( heta ) such that (sin; heta ~=~ -sqrt{1 ~-~ cos^2 ; heta}; ).

3.1.2 We showed that (;cos; heta ~=~ pm,sqrt{1 ~-~ sin^2 ; heta}; ) for all ( heta ). Give an example of an angle ( heta ) such that (cos; heta ~=~ -sqrt{1 ~-~ sin^2 ; heta}; ).

3.1.3 Suppose that you are given a system of two equations of the following form:
[ onumber egin{align*}
A,cos;phi ~ &= ~ B, u_1 ~-~ B u_2 ;cos; heta onumber
A,sin;phi ~ &= ~ B, u_2 ;sin; heta ~.
end{align*}]
Show that (;A ^2 ~=~ B^2 left( u_1^2 ~+~ u_2^2 ~-~ 2 u_1 u_2 ;cos heta ight) ).

For Exercises 4-16, prove the given identity.

3.1.4 (cos; heta ~ an; heta ~=~ sin; heta)

3.1.5 (sin; heta ~ cot; heta ~=~ cos; heta)

3.1.6 (dfrac{ an; heta}{cot; heta} ~=~ an^2 ; heta)

3.1.7 (dfrac{csc; heta}{sin; heta} ~=~ csc^2 ; heta)

3.1.8 (dfrac{cos^2 ; heta}{1 ~+~ sin; heta} ~=~ 1 ~-~ sin; heta)

3.1.9 (dfrac{1 ~-~ 2;cos^2 ; heta}{sin; heta ~ cos; heta} ~=~ an; heta ~-~ cot; heta)

3.1.10 (sin^4 ; heta ~-~ cos^4 ; heta ~=~ sin^2 ; heta ~-~ cos^2 ; heta)

3.1.11 (cos^4 ; heta ~-~ sin^4 ; heta ~=~ 1 ~-~ 2;sin^2 ; heta)

3.1.12 (dfrac{1 ~-~ an; heta}{1 ~+~ an; heta} ~=~
dfrac{cot; heta ~-~ 1}{cot; heta ~+~ 1})

3.1.13 (dfrac{ an; heta ~+~ an;phi}{cot; heta ~+~ cot;phi} ~=~
an; heta ~ an;phi)

3.1.14 (dfrac{sin^2 ; heta}{1 ~-~ sin^2 ; heta} ~=~ an^2 ; heta)

3.1.15 (dfrac{1 ~-~ an^2 ; heta}{1 ~-~ cot^2 ; heta} ~=~ 1 ~-~ sec^2 ; heta)

3.1.16 (sin; heta ~=~ pm,dfrac{ an; heta}{sqrt{1 ~+~
an^2 ; heta}}qquad ) (Hint: Solve for (;sin^2 heta; ) in Exercise 14.)

3.1.17 Sometimes identities can be proved by geometrical methods. For example, to prove the identity in Exercise 16, draw an acute angle ( heta ) in QI and pick the point ((1,y) ) on its terminal side, as in Figure 3.1.2. What must (y ) equal? Use that to prove the identity for acute ( heta ). Explain the adjustment(s) you would need to make in Figure 3.1.2 to prove the identity for ( heta ) in the other quadrants. Does the identity hold if ( heta ) is on either axis?


Figure 3.1.2

3.1.18 Similar to Exercise 16 , find an expression for (cos; heta ) solely in terms of ( an; heta ).

3.1.19 Find an expression for ( an; heta ) solely in terms of (sin; heta ), and one solely in terms of (cos; heta ).

3.1.20 Suppose that a point with coordinates ((x,y)=(a;(cos;psi;-;epsilon),asqrt{1 - epsilon^2}~sin;psi)) is a distance (r>0 ) from the origin, where (a>0 ) and (0 < epsilon < 1 ). Use (;r^2 = x^2 + y^2) to show that (;r = a;(1 ;-; epsilon;cos;psi); ).(Note: These coordinates arise in the study of elliptical orbits of planets.)

3.1.21 Show that each trigonometric function can be put in terms of the sine function.


3.2 Exercises

3.2.1 Verify the addition formulas 3.12 and 3.13 for (A=B=0^circ ).

For Exercises 2 and 3, find the exact values of (sin;(A+B) ), (cos;(A+B) ), and ( an;(A+B) ).

3.2.2 (sin;A = frac{8}{17} ), (cos;A = frac{15}{17} ), (sin;B = frac{24}{25} ),
(cos;B = frac{7}{25})

3.2.3 (sin;A = frac{40}{41} ), (cos;A = frac{9}{41} ), (sin;B = frac{20}{29} ),
(cos;B = frac{21}{29})

3.2.4 Use (75^circ = 45^circ + 30^circ ) to find the exact value of (;sin;75^circ ).

3.2.5 Use (15^circ = 45^circ - 30^circ ) to find the exact value of (; an;15^circ ).

3.2.6 Prove the identity (;sin; heta + cos; heta = sqrt{2};sin;( heta + 45^circ); ). Explain why this shows that
[ onumber
-sqrt{2} ~le~ ;sin; heta ~+~ cos; heta ~le~ sqrt{2}
]
for all angles ( heta ). For which ( heta ) between (0^circ ) and (360^circ ) would (;sin; heta ;+; cos; heta; ) be the largest?

For Exercises 7-14, prove the given identity.

3.2.7 (cos;(A+B+C) ;=; cos;A~cos;B~cos;C ;-;
cos;A~sin;B~sin;C ;-; sin;A~cos;B~sin;C ;-; sin;A~sin;B~cos;C)

3.2.8 ( an;(A+B+C) ~=~ dfrac{ an;A ;+; an;B ;+; an;C ;-;
an;A~ an;B~ an;C}{1 ;-; an;B~ an;C ;-; an;A~ an;C ;-;
an;A~ an;B})

3.2.9 (cot;(A+B) ~=~ dfrac{cot;A~cot;B ;-; 1}{cot;A ;+; cot;B})

3.2.10 (cot;(A-B) ~=~ dfrac{cot;A~cot;B ;+; 1}{cot;B ;-; cot;A})

3.2.11 ( an;( heta + 45^circ) ~=~ dfrac{1 ;+; an; heta}{1 ;-; an; heta})

3.2.12 (dfrac{cos;(A+B)}{sin;A~cos;B} ~=~ cot;A ;-; an;B)

3.2.13 (cot;A ~+~ cot;B ~=~ dfrac{sin;(A+B)}{sin;A~sin;B})

3.2.14 (dfrac{sin;(A-B)}{sin;(A+B)} ~=~
dfrac{cot;B ;-; cot;A}{cot;B ;+; cot;A})

3.2.15 Generalize Exercise 6: For any (a ) and (b ), (-sqrt{a^2 + b^2} ;le; a;sin; heta ;+; b;cos; heta ;le; sqrt{a^2 + b^2}; ) for all ( heta ).

3.2.16 Continuing Example 3.12, use Snell's law to show that the s-polarization transmission Fresnel coefficient
[ ag{3.22}
t_{1;2;s} ~=~ frac{2;n_1 ~cos; heta_1}{n_1 ~cos; heta_1 ~+~ n_2 ~cos; heta_2}
]
can be written as:
[ onumber
t_{1;2;s} ~=~ frac{2;cos; heta_1~sin; heta_2}{sin;( heta_2 + heta_1)}
]

3.2.17 Suppose that two lines with slopes (m_1 ) and (m_2 ), respectively, intersect at an angle ( heta ) and are not perpendicular (i.e. ( heta e 90^circ)), as in the figure on the right. Show that
[ onumber
an; heta ~=~ left| frac{m_1 ~-~ m_2}{1 ~+~ m_1 ; m_2} ight| ~.
]

(Hint: Use Example 1.26 from Section 1.5.)

3.2.18 Use Exercise 17 to find the angle between the lines (y=2x+3 ) and (y=-5x-4 ).

3.2.19 For any triangle ( riangle,ABC ), show that (;cot;A~cot;B ~+~ cot;B~cot;C ~+~
cot;C~cot;A ~=~ 1 ).
(Hint: Use Exercise 9 and (C=180^circ - (A+B) ).)

3.2.20 For any positive angles (A ), (B ), and (C ) such that (A+B+C=90^circ ), show that
[ onumber
an;A~ an;B ~+~ an;B~ an;C ~+~ an;C~ an;A ~=~ 1 ~.
]

3.2.21 Prove the identity (;sin;(A+B)~cos;B ~-~ cos;(A+B)~sin;B ~=~ sin;A ). Note that the right side depends only on (A ), while the left side depends on both (A ) and (B ).

3.2.22 A line segment of length (r > 0 ) from the origin to the point ((x,y) ) makes an angle (alpha ) with the positive (x)-axis, so that ((x,y) = (r;cos;alpha,r;sin;alpha) ), as in the figure below. What are the endpoint's new coordinates ((x',y') ) after a counterclockwise rotation by an angle (eta;)? Your answer should be in terms of (r ), (alpha ), and (eta ).


3.3 Exercises

For Exercises 1-8, prove the given identity.

3.3.1 (cos;3 heta ~=~ 4;cos^3 ; heta ~-~ 3;cos; heta)

3.3.2 ( an; frac{1}{2} heta ~=~ csc; heta ~-~ cot; heta)

3.3.3 (dfrac{sin;2 heta}{sin; heta} ~-~ dfrac{cos;2 heta}{cos; heta} ~=~ sec; heta)

3.3.4 (dfrac{sin;3 heta}{sin; heta} ~-~ dfrac{cos;3 heta}{cos; heta} ~=~ 2)

3.3.5 ( an;2 heta ~=~ dfrac{2}{cot; heta ;-; an; heta})

3.3.6 ( an;3 heta ~=~ dfrac{3; an; heta ;-; an^3 ; heta}{1 ;-; 3; an^2 ; heta})

3.3.7 ( an^2 ; frac{1}{2} heta ~=~ dfrac{ an; heta ;-; sin; heta}{ an; heta ;+; sin; heta})

3.3.8 (dfrac{cos^2 ;psi}{cos^2 ; heta} ~=~ dfrac{1 ;+; cos;2psi}{1 ;+;
cos;2 heta})

3.3.9 Some trigonometry textbooks used to claim incorrectly that (;sin; heta ~+~ cos; heta ~=~ sqrt{1 ;+; sin;2 heta} ) was an identity. Give an example of a specific angle ( heta ) that would make that equation false. Is (;sin; heta ~+~ cos; heta ~=~ pm;sqrt{1 ;+; sin;2 heta} ) an identity? Justify your answer.

3.3.10 Fill out the rest of the table below for the angles (0^circ < heta < 720^circ ) in increments of (90^circ ), showing ( heta ), ( frac{1}{2} heta ), and the signs ((+) or (-)) of (sin; heta ) and ( an; frac{1}{2} heta ).

3.3.11 In general, what is the largest value that (;sin; heta~cos; heta; ) can take? Justify your answer.

For Exercises 12-17, prove the given identity for any right triangle ( riangle,ABC ) with (C=90^circ ).

3.3.12 (sin;(A-B) ~=~ cos;2B)

3.3.13 (cos;(A-B) ~=~ sin;2A)

3.3.14 (sin;2A ~=~ dfrac{2;ab}{c^2})

3.3.15 (cos;2A ~=~ dfrac{b^2 - a^2}{c^2})

3.3.16 ( an;2A ~=~ dfrac{2;ab}{b^2 - a^2})

3.3.17 ( an; frac{1}{2}A ~=~ dfrac{c - b}{a} ~=~ dfrac{a}{c + b})

3.3.18 Continuing Exercise 20 from Section 3.1, it can be shown that
[ egin{align*}
r;(1 ;-; cos; heta) ~&=~ a;(1 ;+; epsilon),(1 ;-; cos;psi) ~,~ ext{and}
r;(1 ;+; cos; heta) ~&=~ a;(1 ;-; epsilon),(1 ;+; cos;psi) ~,
end{align*}]
where ( heta ) and (psi ) are always in the same quadrant. Show that (; an; frac{1}{2} heta ~=~ sqrt{frac{1 ;+; epsilon}{1 ;-; epsilon}}~ an; frac{1}{2}psi; ).


3.4 Exercises

3.4.1 Prove formula 3.38.

3.4.2 Prove formula 3.39.

3.4.3 Prove formula 3.40.

3.4.4 Prove formula 3.41.

3.4.5 Prove formula 3.42.

3.4.6 Prove formula 3.44.

3.4.7 Prove Mollweide's second equation: For any triangle ( riangle,ABC ), (~dfrac{a+b}{c} ~=~ dfrac{cos; frac{1}{2}(A-B)}{sin; frac{1}{2}C}).

3.4.8 Continuing Example 3.21, use Snell's law to show that the p-polarization reflection Fresnel coefficient
[ ag{3.46}
r_{1;2;p} ~=~ frac{n_2 ~cos; heta_1 ~-~ n_1 ~cos; heta_2}{n_2 ~cos; heta_1 ~+~
n_1 ~cos; heta_2}
]
can be written as:
[
r_{1;2;p} ~=~ frac{ an;( heta_1 - heta_2)}{ an;( heta_1 + heta_2)}
]
3.4.9 There is a more general form for the instantaneous power (p(t) = v(t);i(t) ) in an electrical circuit than the one in Example 3.22. The voltage (v(t) ) and current (i(t) ) can be given by
[ egin{align*}
v(t) ~&=~ V_m ;cos;(omega t + heta)~,
i(t) ~&=~ I_m ;cos;(omega t + phi)~,
end{align*}]
where ( heta ) is called the phase angle. Show that (p(t) ) can be written as
[
p(t) ~=~ frac{1}{2},V_m ; I_m ;cos;( heta - phi) ~+~
frac{1}{2},V_m ; I_m ;cos;(2omega t + heta + phi) ~.
]

For Exercises 10-15, prove the given identity or inequality for any triangle ( riangle,ABC ).

3.4.10 (sin;A ;+; sin;B ;+; sin;C ~=~
4;cos; frac{1}{2}A~cos; frac{1}{2}B~cos; frac{1}{2}C) (Hint: Mimic Example 3.18 using ((sin;A ;+; sin;B) ;+; (sin;C ;-; sin;(A+B+C)) ).)

3.4.11 (cos;A ;+; cos;(B-C) ~=~ 2;sin;B~sin;C)

3.4.12 (sin;2A ;+; sin;2B ;+; sin;2C ~=~ 4;sin;A~sin;B~sin;C) (Hints: Group (sin;2B ) and (sin;2C ) together, use the double-angle formula for (sin;2A ), use Exercise 11.)

3.4.13 (dfrac{a-b}{a+b} ~=~ dfrac{sin;A ;-; sin;B}{sin;A ;+; sin;B})

3.4.14 (cos; frac{1}{2}A ~=~ sqrt{dfrac{s;(s-a)}{bc}}~~ ) and (~~sin; frac{1}{2}A ~=~ sqrt{dfrac{(s-b);(s-c)}{bc}}; ), ;where (s= frac{1}{2}(a+b+c)$) (Hint: Use the Law of Cosines to show that (2bc;(1 + cos;A) ~=~ 4s;(s-a) ).)

3.4.15 ( frac{1}{2};(sin;A ;+; sin;B) ~le~
sin; frac{1}{2}(A+B)) (Hint: Show that (sin; frac{1}{2}(A+B) ;-;
frac{1}{2};(sin;A ;+; sin;B) ;ge; 0 ).)

3.4.16 In Example 3.20, which angles (A ), (B ), (C ) give the maximum value of (cos;A ;+; cos;B ;+; cos;C;)?


Mathematics 8 (MYP 3) (3rd edition)

Mathematics 8 (MYP 3) third edition has been designed and written for the International Baccalaureate Middle Years Programme (IB MYP) Mathematics framework, providing complete coverage of the content and expectations outlined.

Discussions, Activities, Investigations, and Research exercises are used throughout the chapters to develop conceptual understanding. Material is presented in a clear, easy-to-follow style to aid comprehension and retention, especially for English Language Learners. Each chapter ends with extensive review sets and an online multiple-choice quiz.

The associated digital Snowflake subscription supports the textbook content with interactive and engaging resources for students and educators.

The Global Context projects highlight the use of mathematics in understanding history, culture, science, society, and environment. We have aimed to provide a diversity of topics and styles to create interest for all students and illustrate the real-world application of mathematics.

We have developed this book in consultation with experienced teachers of IB Mathematics internationally but independent of the International Baccalaureate Organisation (IBO). It is not endorsed by the IBO.

We have endeavoured to publish a stimulating and thorough textbook and digital resource to develop and encourage student understanding and nurturing an appreciation of mathematics.

Year Published: 2021
Page Count: 496
ISBN: 978-1-922416-32-2 (9781922416322)
Online ISBN: 978-1-922416-33-9 (9781922416339)

Mathematics 8 (MYP 3) (3rd edition)

GLOBAL CONTEXTS 9
1 NUMBER 11
A Operations with negative numbers 13
B Exponent notation 15
C Factors 17
D Prime and composite numbers 17
E Highest common factor 19
F Multiples 20
G Order of operations 22
H Problem solving 25
Review set 1A 27
Review set 1B 28
2 SETS AND VENN DIAGRAMS 29
A Sets 30
B Complement of a set 32
C Intersection and union 34
D Venn diagrams 36
E Numbers in regions 40
F Problem solving with Venn diagrams 42
Review set 2A 43
Review set 2B 45
3 REAL NUMBERS 47
A Fractions 48
B Equal fractions 51
C Adding and subtracting fractions 52
D Multiplying fractions 54
E Dividing fractions 56
F Decimal numbers 57
G Rounding decimal numbers 58
H Adding and subtracting decimal numbers 59
I Multiplying and dividing by powers of $10$ 60
J Multiplying decimal numbers 61
K Dividing decimal numbers 62
L Square roots 64
M Cube roots 65
N Rational numbers 66
O Irrational numbers 69
Review set 3A 71
Review set 3B 73
4 ALGEBRAIC EXPRESSIONS 75
A Product notation 76
B Exponent notation 77
C Writing expressions 78
D Generalising arithmetic 80
E Algebraic substitution 81
F The language of algebra 84
G Collecting like terms 86
H Algebraic products 87
I Algebraic fractions 88
J Multiplying algebraic fractions 89
K Dividing algebraic fractions 90
L Algebraic common factors 91
Review set 4A 92
Review set 4B 93
5 PERCENTAGE 95
A Converting percentages into decimals and fractions 96
B Converting decimals and fractions into percentages 98
C Expressing one quantity as a percentage of another 99
D Finding a percentage of a quantity 100
E The unitary method for percentages 101
F Percentage increase or decrease 103
G Finding a percentage change 106
H Finding the original amount 108
I Profit and loss 109
J Discount 111
K VAT and GST 113
Review set 5A 114
Review set 5B 115
6 LAWS OF ALGEBRA 117
A Exponent laws 118
B Expansion laws 122
C The zero exponent law 124
D The negative exponent law 126
E The distributive law 128
F Factorisation 132
Review set 6A 134
Review set 6B 135
7 EQUATIONS 137
A Solutions of an equation 138
B Maintaining balance 140
C Inverse operations 142
D Algebraic flowcharts 145
E Solving equations 146
F Equations with a repeated unknown 150
G Power equations 153
Review set 7A 156
Review set 7B 157
8 LINES AND ANGLES 159
A Angles 160
B Parallel and perpendicular lines 162
C Angle properties 163
D Lines cut by a transversal 164
Review set 8A 166
Review set 8B 167
9 PLANE GEOMETRY 169
A Circles 170
B Triangles 172
C Triangle theorems 174
D Isosceles triangles 177
E Quadrilaterals 181
F Angle sum of a quadrilateral 184
G Angle sum of an $n$-sided polygon 186
Review set 9A 189
Review set 9B 191
10 ALGEBRA: FORMULAE 193
A Number crunching machines 194
B Finding the formula 196
C Substituting into formulae 198
D Geometric patterns 201
E Practical problems 204
Review set 10A 206
Review set 10B 208
11 MEASUREMENT: LENGTH AND AREA 211
A Length 212
B Perimeter 215
C Circumference 218
D Area 222
E Area formulae 224
F The area of a circle 228
G Areas of composite figures 231
Review set 11A 236
Review set 11B 238
12 MEASUREMENT: SURFACE AREA, VOLUME, AND CAPACITY 241
A Surface area 242
B Surface area of a cylinder 244
C Surface area of a sphere 247
D Volume 248
E Volume of a solid of uniform cross-section 250
F Volume of a tapered solid 254
G Volume of a sphere 256
H Capacity 257
I Connecting volume and capacity 258
Review set 12A 260
Review set 12B 261
13 TIME 263
A Units of time 264
B Time calculations 268
C $24$-hour time 271
D Time zones 273
Review set 13A 276
Review set 13B 277
14 COORDINATE GEOMETRY 279
A The Cartesian plane 280
B Straight lines 284
C Gradient 288
D The gradient-intercept form of a line 292
E Graphing a line from its gradient-intercept form 294
F The $x$-intercept of a line 295
G Graphing a line from its axes intercepts 297
H Finding the equation from the graph of a line 297
Review set 14A 299
Review set 14B 301
15 RATIO 303
A Ratio 304
B Equal ratios 306
C Lowest terms 307
D Proportions 310
E Using ratios to divide quantities 311
F Scale diagrams 313
Review set 15A 318
Review set 15B 319
16 RATES AND LINE GRAPHS 321
A Rates 322
B Speed 325
C Density 327
D Converting rates 330
E Line graphs 332
Review set 16A 336
Review set 16B 337
17 PROBABILITY 339
A Probability 340
B Sample space 342
C Theoretical probability 343
D Independent events 347
E Experimental probability 350
F Probabilities from tabled data 352
G Probabilities from two-way tables 353
H Probabilities from Venn diagrams 356
I Expectation 357
Review set 17A 359
Review set 17B 361
18 STATISTICS 363
A Data collection 364
B Categorical data 367
C Numerical data 372
D Grouped data 374
E Stem-and-leaf plots 376
F Measures of centre and spread 378
G Measures of centre and spread from a frequency table 383
Review set 18A 385
Review set 18B 388
19 CONGRUENCE AND SIMILARITY 391
A Congruence 392
B Congruent triangles 394
C Proof using congruence 400
D Enlargements and reductions 402
E Similarity 404
F Similar triangles 408
G Problem solving 411
Review set 19A 414
Review set 19B 416
20 PYTHAGORAS' THEOREM 419
A Pythagoras' theorem 421
B Problem solving 425
C The converse of Pythagoras' theorem 427
Review set 20A 429
Review set 20B 430
21 PROBLEM SOLVING 431
A Writing problems as equations 432
B Problem solving with algebra 433
C Solution by search 436
D Solution by working backwards 438
E Miscellaneous problems 440
F Lateral thinking 442
Review set 21A 448
Review set 21B 449
ANSWERS 451
INDEX 494

Authors

Michael Haese

Michael completed a Bachelor of Science at the University of Adelaide, majoring in Infection and Immunity, and Applied Mathematics. He studied laminar heat flow as part of his Honours in Applied Mathematics, and finished a PhD in high speed fluid flows in 2001. He has been the principal editor for Haese Mathematics since 2008.

What motivates you to write mathematics books?

My passion is for education as a whole, rather than just mathematics. In Australia I think it is too easy to take education for granted, because it is seen as a right but with too little appreciation for the responsibility that goes with it. But the more I travel to places where access to education is limited, the more I see children who treat it as a privilege, and the greater the difference it makes in their lives. But as far as mathematics goes, I grew up with mathematics textbooks in pieces on the kitchen table, and so I guess it continues a tradition.

What do you aim to achieve in writing?

  • I want to write to the student directly, so they can learn as much as possible from the text directly. Their book is there even when their teacher isn't.
  • I therefore want to write using language which is easy to understand. Sure, mathematics has its big words, and these are important and we always use them. But the words around them should be as simple as possible, so the meaning of the terms can be properly explained to ESL (English as a Second Language) students.
  • I want to make the mathematics more alive and real, not by putting it in contrived "real-world" contexts which are actually over-simplified and fake, but rather through its history and its relationship with other subjects.

What interests you outside mathematics?

Lots of things! Horses, show jumping and course design, alpacas, badminton, running, art, history, faith, reading, hiking, photography .

Mark Humphries

Mark has a Bachelor of Science (Honours), majoring in Pure Mathematics, and a Bachelor of Economics, both of which were completed at the University of Adelaide. He studied public key cryptography for his Honours in Pure Mathematics. He started with the company in 2006, and is currently the writing manager for Haese Mathematics.

What got you interested in mathematics? How did that lead to working at Haese Mathematics?

I have always enjoyed the structure and style of mathematics. It has a precision that I enjoy. I spend an inordinate amount of my leisure time reading about mathematics, in fact! To be fair, I tend to do more reading about the history of mathematics and how various mathematical and logic puzzles work, so it is somewhat different from what I do at work.

How did I end up at Haese Mathematics?

I was undertaking a PhD, and I realised that what I really wanted to do was put my knowledge to use. I wanted to pass on to others all this interesting stuff about mathematics. I emailed Haese Mathematics (Haese and Harris Publications as they were known back then), stating that I was interested in working for them. As it happened, their success with the first series of International Baccalaureate books meant that they were looking to hire more people at the time. I consider myself quite lucky!

What are some interesting things that you get to do at work?

On an everyday basis, it&rsquos a challenge (but a fun one!) to devise interesting questions for the books. I want students to have questions that pique their curiosity and get them thinking about mathematics in a different way. I prefer to write questions that require students to demonstrate that they understand a concept, rather than relying on rote memorisation.

When a new or revised syllabus is released for a curriculum that we write for, a lot of work goes into devising a structure for the book that addresses the syllabus. The process of identifying what concepts need to be taught, organising those concepts into an order that makes sense from a teaching standpoint, and finally sourcing and writing the material that addresses those concepts is very involved &ndash but so rewarding when you hold the finished product in your hands, straight from the printer.

What interests you outside mathematics?

Apart from the aforementioned recreational mathematics activities, I play a little guitar, and I enjoy playing badminton and basketball on a social level.


Gamma Function

Symbolic Computing

It is recommended that symbolic computations of exponential integrals be carried out using the function defined as E 1 ( x ) . In both maple and mathematica a command to evaluate E n ( x ) , Eq. (9.52) , is available. In maple , it is Ei(n,x) in mathematica it is ExpIntegralE[n,x] . The function E 1 can therefore be obtained from these commands by setting n = 1 . Both languages also contain quantities corresponding to Ei ( x ) , namely Ei(x) and ExpIntegralEi[x] . These are not synonymous with E 1 ( x ) and cannot be used when E 1 is intended.

Example 9.6.1 Symbolic Computing, Exponential Integral

The function you get depends on whether Ei is called with one or with two arguments.

Here ExpIntegralE and ExpIntegralEi must be called with the correct numbers of arguments. ▪


3.E: Identities (Exercises) - Mathematics

Percentiles are measures that divide a group of data into 100 parts.
Percentiles are values that split your data into percentages in the same way that quartiles split data into quartes. Each percentile is referred to by the percentage with which it splits the data. so 10th percentile i sthe value that is 10% os the way through the data.

In general, the kth percentile is the value that is k% of the way through the data. It's usually denoted by Pk

  • P is the percentile of interest
  • i is the percentile location
  • N is the number in the data set

Step 3. Determine the location by either (a) or (b)

a. If i is a whole number, the Pth percentile is the average of the value at the ith location and the value at the (i+1)st location.
b. If i is not a whole number, the PTh percentile value is located at the whole number part of i+1

Determine the 30th percentile of the following eight numbers 1 2 4 3 5 3 5 2 6

Step 1. Organize the data into an ascending-order array: 1 2 2 3 3 4 5 5 6
Step 2. Calculate the percentile location:
Step 3. Determine the location: Because i is not a whole number, step 3(b) is used. The value of i+1 is 2.4+1, or 3.4. The whole number part of 3.4 is 3. The 30th percentile is located as the third value. The third value is 2, so 2 is the 30th percentile.

  • L is the lower limit of the class containing Pk
  • f is the frequency of the class containing Pk
  • h is the width of the class containing Pk
  • C is the cumulative frequency of the class preceding the class containing Pk

Here, the cumutative frequency just greater than is the class containing Pk (K=1,2. 99)


Algebraic Identities

An algebraic identity is an equality that holds for any values of its variables.

Since an identity holds for all values of its variables, it is possible to substitute instances of one side of the equality with the other side of the equality. For example, because of the identity above, we can replace any instance of ( x + y ) 2 (x+y)^2 ( x + y ) 2 with x 2 + 2 x y + y 2 x^2 + 2xy + y^2 x 2 + 2 x y + y 2 and vice versa.

Clever use of identities offers shortcuts to many problems by making the algebra easier to manipulate. Below are lists of some common algebraic identities.

Contents


Expand the Algebraic Expressions using Identities

Employ this batch of printable worksheets to enhance your skills in applying algebraic identities to expand algebraic expressions. Each section offers two levels of difficulty other than cubic expressions. Expand the algebraic expressions in the standard form (a+b) 2 , (a-b) 2 , (a+b) (a-b), (x+a) (x+b), (a+b+c) 2 , quadratic expressions, cubic expressions and much more. The exercises are curated for 8th grade and high school students. Procure some of these worksheets for free!

Expand the algebraic expressions involving single and multivariables by using either (a+b) 2 or (a-b) 2 in these grade 8 worksheets.

Level: Easy, Moderate (3 worksheets each)

This batch of worksheets represents algebraic expressions as a product of two binomials. Apply the formula (a+b) (a-b) = a 2 - b 2 , to expand each algebraic expression.

Level: Easy, Moderate (3 worksheets each)

Use our resources for extensive practice to determine the product of two binomials, whose first terms are same and second terms are different. Compare the expression with the identity (x+a) (x+b) = x 2 + (a+b) x + ab to expand the given algebraic expression.

Level: Easy, Moderate (3 worksheets each)

Give high school students a jump start on the expansion of the square of a trinomial with these printable worksheets. Build up the skills in applying the identity (a+b+c) 2 = a 2 + b 2 + c 2 + 2 (ab + bc + ca) to expand the expressions.

Level: Easy, Moderate (3 worksheets each)

Strengthen your algebraic skills in using the identities (a+b) 3 and (a-b) 3 to expand the set of algebraic expressions enclosed in these amazing series of pdf worksheets.

Algebraic expressions featured in this compilation of worksheets are in the form (a+b) (a 2 - ab + b 2 ) and (a-b) (a 2 + ab + b 2 ). Apply the sum of cubes or difference of cubes identity to determine the product.

These printable algebra worksheets feature algebraic expressions in the form of square identities. Look at the expression and apply suitable identity to expand the expression.

Level: Easy, Moderate (3 worksheets each)

Explore this batch of challenging worksheet pdfs to learn how to apply cube identities to expand the algebraic expressions. It provides students with a good foundation in learning algebraic identities.


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Chapter 02: Sets, Functions and Groups
Chapter 03: Matrices and Determinants
Chapter 04: Quadratic Equations
Chapter 05: Partial Fractions
Chapter 06: Sequences and Series
Chapter 07: Permutation , Combination and Probability
Chapter 08: Mathematical Induction and Binomial Theorem
Chapter 09: Fundamentals of Trigonometry
Chapter 10: Trigonometric Identities
Chapter 11: Trigonometric Functions and their Graphs
Chapter 12: Application of Trigonometry
Chapter 13: Inverse Trigonometric Functions
Chapter 14: Solutions of Trigonometric Equation

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Discrete Mathematics

Discrete Mathematics: An Open Introduction is a free, open source textbook appropriate for a first or second year undergraduate course for math majors, especially those who will go on to teach. Since Spring 2013, the book has been used as the primary textbook or a supplemental resource at more than 75 colleges and universities around the world (see the partial adoptions list). The text is endorsed by the American Institute of Mathematics' Open Textbook Initiative and is well reviewed on the Open Textbook Library.

This 3rd edition brings many improvements, including nearly 100 new exercises, a new section on trees in the graph theory chapter, and improved exposition throughout. Previous editions will continue to be available indefinitely. A few times a year, the text is updated with a new "printing" to correct errors. See the errata list for more information.

New for Fall 2019: Online homework sets are available through Edfinity or as WeBWorK sets from the author. Additional exercises have been added since Spring 2020.

Please contact the author with feedback and suggestions, or if you are decide to use the book in a course you are teaching.

Get the book

The entire book is available for free as an interactive online ebook. This should work well on all screen sizes, including smart phones. Hints and solutions to examples and exercises are hidden but easily revealed by clicking on their links. Some exercises also allow you to enter and check your work, so you can try multiple times without spoiling the answer.

For offline use, a free pdf version, suitable for reading on a tablet or computer, is available for download. This should be searchable and easy to navigate using embedded links. Hints and solutions (when available) can be accessed by clicking on the exercise number, and clicking on the number of the hint or solution will bring you back to the exercise.

If you prefer a physical copy, an inexpensive print version of the text is available on Amazon. This should be cheaper than printing the entire book and binding it yourself. Page numbers match the pdf version.

PreTeXt (and LaTeX) source

The source files for this book are available on GitHub.

Instructor resources

If you are using the book in a class you are teaching, instructor resources are available by request. Just contact the author. You can also request WeBWorK homework sets if you have access to a WeBWorK server (otherwise, consider using the reasonably priced Edfinity).

About the book

The text began as a set of lecture notes for the discrete mathematics course at the University of Northern Colorado. This course serves both as an introduction to topics in discrete math and as the "introduction to proofs" course for math majors. The course is usually taught with a large amount of student inquiry, and this text is written to help facilitate this.

Four main topics are covered: counting, sequences, logic, and graph theory. Along the way, proofs are introduced, including proofs by contradiction, proofs by induction, and combinatorial proofs. An introductory chapter covering mathematical statements, sets, and functions helps students gain familiarity with the language of mathematics, and two additional topics (generating functions and number theory) are also included.

  • 473 exercises, including 275 with solutions and another 109 with hints. Exercises range from easy to quite involved, with many problems suitable for homework.
  • Investigate! activities throughout the text to support active, inquiry based learning.
  • A full index and list of symbols.
  • Consistent and helpful page layout and formatting (i.e., examples are easy to identify, important definitions and theorems in boxes, etc.).

About the author

Oscar Levin is an associate professor at the University of Northern Colorado. He has taught mathematics at the college level for over 10 years and received multiple teaching awards. He received his Ph.D. in mathematical logic from the University of Connecticut in 2009.

License

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Discrete Mathematics: An Open Introduction by Oscar Levin is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. You are free to download, use, print, and even sell this work as you wish to. You can also modify the text as much as you like (create a custom edition for your students, for example), as long as you attribute the parts of the text you use to the author.

If you are interested in using parts of the book combined with another text with a similar but different license (GFDL, for example), please reach out to get permission to modify the license.


Sometimes we can hear about combinatorial proofs of a problem and sometimes we hear about proofs based upon formal or symbolic methods. Combinatorial proofs typically search for bijections between known finite sets and the objects we like to count and going this way we try to get a deeper understanding about the underlying structure of these objects. On the other hand symbolic methods are based upon different types of generating functions. With the help of these functions many counting problems can be easily solved by rather simple algebraic methods using formal, finite operations and without considering limits or other analytic means.

One classic providing an enormous amount of combinatorial proofs is Richard P. Stanleys Enumerative Combinatorics Volume $1$ and $2$ . You was asking for more than one proof of a structure and you will be satisfied. E.g. example $6.19$ of Volume $2$ gives you $66$ different sets of the famous Catalan Numbers $frac<1>inom<2n>$ at hand. You will find there many wonderful examples with combinatorial proofs.

Some other prior classic is Advanced Combinatorics $(1974)$ from Louis Comtet. This book is also a great guide through the landscape of combinatorics. It contains many particular problems with combinatorial proofs.

These two books are my recommendation for combinatorial proofs. I'd like to add some more hints to complete the (my) picture:

The book Combinatorial Identities from John Riordan ( $1968$ ) is a wonderful classic with thousands of binomial identities which are systematically organised. But it does not typically provide combinatorial proofs. It's a great reference to search for different classes of combinatorial identities.

If you also consider to have a look at the other, formal side then H.Wilf's book Generatingfunctionology is the perfect, easily accessible starter to see the power of formal series.

A great book, presumably playing in the same league as Stanleys Enumerative Combinatorics is Analytic Combinatorics from Philippe Flajolet and Robert Sedgewick. Here you will not only find a definitive reference of symbolic methods in Combinatorics (first part of the book), but also how the great power of complex analysis can be used to get information about asymptotic behaviour, singularity analysis of generating functions and many other beautiful things.

The guiding theme for all these references is: Read it, analyse it (at least partly) and have fun :-)


3.E: Identities (Exercises) - Mathematics

NCERT Solutions for Class 8 Maths consists of the chapter-wise solutions of all the problems provided in the NCERT textbook for Class 8 Mathematics. GeeksforGeeks has created a detailed chapter-wise solution for the NCERT book of class 8 that contains problems on various topics like Rational Numbers, Linear Equations, Quadrilaterals, Data Handling, and many more. Each chapter in this solution thoroughly covers every exercise along with a detailed step-by-step explanation of the solutions.

Chapter 1: Rational Numbers

The chapter Rational numbers mainly discuss the characteristics of all the real numbers, integers, whole numbers, rational numbers, and natural numbers. This chapter consists of two exercises only in which the problems in Exercise 1.1 are related to the properties of the rational numbers (closure, commutativity, associativity, etc). However, in Exercise 1.2 the problems are related to the advanced concepts of the rational number like the representation of rational numbers on a number line and to determine any numbers of rational number between any two rational numbers.

Chapter 2: Linear Equations in One Variable

The linear equations in one variable deal with the expression defined in one variable only and its algebraic operations. This chapter contains six different exercises that have problems based on the linear equations in one variable and its application. Exercises 2.1, 2.2, 2.3, and 2.4 are designed to determine the solution of the linear equation. However, Exercises 2.5 and 2.6 are based on the topic e quations reducible to the linear form.

Chapter 3: Understanding Quadrilaterals

This chapter covers all types of quadrilaterals such as polygonal shapes like square, rectangle, triangle, pentagon, hexagon, etc. In total, this chapter contains four exercises in which Exercise 3.1 covers the problems on the definition of various polygons and their properties, Exercise 3.2 is based on the concept of the Angle sum property of a polygon. However, Exercise 3.3 covers the elements and the properties of quadrilaterals like trapezium, kite, and parallelogram, and Exercise 3.4 is designed to learn some special types of parallelogram like square, rectangle, and rhombus.

Chapter 4: Practical Geometry

The chapter practical geometry helps to learn the construction of quadrilateral when different parameters of it are known. This chapter contains a total of five exercises only in which Exercise 4.1 covers the problem for the case when the lengths of four sides and a diagonal are given. Similarly, E xercises 4.2, 4.3 , and 4.4 are based on the topics when two diagonals and three sides are known, two adjacent sides and three angles are given and three sides and two included angles are provided. However, Exercise 4.5 contains problems based on some special cases.

Chapter 5: Data Handling

Data handling is a method of organizing data or information systematically using diagrams like bar graphs, pictographs, pie charts, and histograms. This chapter consists of only three exercises. Exercise 5.1 is based on the basic concept of representing, organizing, and grouping the data provided while Exercise 5.2 helps to make a pie chart for the given data. Moreover, Exercise 5.3 coves the topic that helps to understand the basic concept of probability.

Chapter 6: Squares and Square Roots

As the name of the chapter says, Squares and square roots this chapter gives the knowledge of the concept to determine the squares and square root of a number. The different properties and the pattern followed to find a square number are discussed in four exercises. Exercises 6.1 and 6.2 are based on the basic idea of the square numbers and different ways to determine them. Though Exercises 6.4 and 6.5 are focused on the concept of the determination of the square root of a number.

Chapter 7: Cubes and Cube roots

Again, as the title of the chapter suggest that Cubes and Cube roots this chapter helps to understand the concept to determine the cubes and cube root of a number. The different patterns followed to find a cube and cube root of a number are discussed in only two exercises. Exercises 7.1 contains the problem to determine whether the given number is a perfect cube or not. And Exercises 7.2 focused on the idea of the cube root and the determination of the cube root of a number.

Chapter 8: Comparing Quantities

This chapter gives a basic understanding of the topics such as increased and decreased percentage, market price, selling price, cost price, discount, and discount price, profit or loss, interest, etc. Total there are three exercises in this chapter, Exercise 8.1 based on the topics ratios and percentage, and Exercises 8.2 and 8.3 covers a wide range of concepts such as percentage, profit or loss, tax, and compound interest.

Chapter 9: Algebraic Expressions and Identities

Chapter 10: Visualising Solid Shapes

This chapter provides the understanding of different solids shapes when visualized in different dimensions and various terms used to describe their properties. The chapter explains this in three different exercises. Here exercises 10.1 and 10.2 are based on the concept of the visualisation of different solid shapes at different positions and the mapping spaces around the observer. Though, Exercise 10.3 discussed the terms like faces, edges, vertices, and relation between them, related to a solid shape.

Chapter 11: Mensuration

Mensuration is the chapter that deals with the measurement or the calculations related to determine the area, perimeter, volume of various geometrical figures like square, cube, rectangle, cuboid, cylinder, and triangle, etc. This chapter consists of only four exercises in which Exercises 11.1 and 11.2 deal with problems related to the areas of different geometrical shapes, combination of shapes, and every-day life examples. However, Exercises 11.3 and 11.4 discussed the terminology related to 3-Dimensional shapes.

Chapter 12: Exponents and Powers

The chapter Exponents and powers cover the primary concepts such as laws of exponents and their applications. This chapter consists of only two exercises, Exercise 12.1 is specifically based on the laws of exponents, and Exercise 12.2 deals with the problems using the applications of power to write large numbers in exponents and vice-versa.

Chapter 13: Direct and Inverse Proportions

This chapter gives a detailed explanation of inverse and direct proportions through problems discussed in two exercises. In which Exercise 13.1 contains problems to determine the direct proportions between any quantity and Exercise 13.2 deals with the questions from the indirect inverse.

Chapter 14: Factorisation

This chapter comprises the problems on the factors of natural numbers and algebraic expressions, factorisation by regrouping terms, factorisation using identities, division of algebraic expressions. The chapter includes four exercises out of which exercises 14.1 and 14.2 are based on the topic factorisations and their application while exercises 14.3 and 14.4 emphasize the division of algebraic expressions.

Chapter 15: Introduction to Graphs

This chapter is all about the basic understanding of the graphs, kinds of graphs, etc. It is mainly explained using three exercises, Exercise 15.1 deals with problems from introduction to graphs and terminology related to it while, problems in Exercises 15.2 and 15.3 provided emphasis on the construction of different types of graphs and their applications.

Chapter 16: Playing with Numbers

All the above-mentioned chapters basically helped to learn about various kinds of numbers and their different properties likewise in this chapter the concept of numbers is discussed in a more general way. This chapter includes two exercises only, Exercise 16.1 and 16.2 which contains fun activities, puzzles, etc. such as divisibility tests to determine any missing number in a series of numbers.


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