SingaporeMath.com Inc

Trigonometry, a work in the collection of the Gelfand School Program, is the result of a collaboration between two experienced pre-college teachers, one of whom, I.M Gelfand, is considered to be among our most distinguished living mathematicians.  His impact on generations of young people, some now mathematicians of renown, continues to be remarkable.

Trigonometry covers all the basics of the subject through beautiful illustrations and examples.  The definitions of the trigonometric functions are geometrically motivated.  Geometric relationships are rewritten in trigonometric form and extended.  The text then makes a transition to the study of algebraic and analytic properties of trigonometric functions, in a way that provides a solid foundation for more advanced mathematical discussions.  Throughout, the treatment stimulates the reader to think of mathematics as a unified subject. 

Like otherI.M. Gelfand treasures in the program - Algebra, Functions and Graphs, and The Method of Coordinates - Trigonometry is written in engaging style, and approaches the material in a unique fashion that will motivate students and teachers alike.

Above are extracts from the book Trigonometry (with permission of the publishers).

 Trigonometry by Gelfand and Saul

0. Trigonometry

                1      What is new about trigonometry?

      2      Right triangles

      3      The Pythagorean theorem        

                4      Our best friends (among right triangles)

      5      Our next best friends (among right triangles)

                6      Some standard notation 

 

1. Trigonometry Ratios in a Triangle

                1        Definition of sin 

      2        Find the hidden sine

                3        The cosine ratio

                4        A relation between the sine and the cosine

                5        A bit of notation

                6        Another relation between the sine and the cosine

                7        Our next best friends (and the sine ratio)

      8        What is the value of sin 90º    

                9        An exploration: How large can the sum be?

              10        More explanation: How large can the product be?

              11        More names for ratios    

 

2. Relations among Trigonometric Ratios

                1        The sine and its relatives

      2        Algebra or geometry?

                3        A remark about names

      4        An identity crisis?

                5        Identities with secant and cosecant

      6        A lemma

      7        Some inequalities

      8        Calculations and tables

      9        Getting the degree measure of an angle from its sine

     10       Solving right triangles

     11       Shadows

     12       Another approach to the sine ration

   

3. Relationships in a Triangle

                1        Geometry of the triangle

      2        The congruence theorems and trigonometry

      3        Sines and altitudes

      4        Obtuse triangles

      5        The Law of Sines

      6        The circumradius

      7        Area of a triangle

      8        Two remarks

      9        Law of cosines

                          

4. Angles and Rotations

                1        Measuring rotations

      2        Rotation and angles

      3        Trigonometric functions for all angles

      4        Calculations with angles of rotations

      5        Odd and even functions

              

5. Radian Measure

                1        Radian measure for angles and rotations

      2        Radian measure and distance

      3        Interlude: How to explain radian measure to your brother and sister     

      4        Radian measure and calculators

      5        An important graph

      6        Two small miracles    

 

6. The Addition Formulas

                1        More identities

      2        The addition formulas

                3        Proofs of the addition formulas

      4        A first beautiful proof

                5        A second beautiful proof

 

7. Trigonometric Identities

                1        Extending the identities

      2        The Principle of Analytic Continuation Higher mathematics to the rescue

      3        Back to our identities

      4        A formula for tan (x+ß)

      5        Double the angle

      6        Triple the angle

      7        Derivation of the formulas for sin x/2 and cos x/2   

      8        Another formula for tan x/2

      9        Products to sums

     10       Sums to products

                  

8. Graphs of Trigonometric Functions

                1        Graphing the basic sine curve

      2        The period of the function y = sin x

      3        Periods of other sinusoidal curves

      4        The amplitude of a sinusoidal curve

                5        Shifting the sine

      6        Shifting and stretching

      7        Some special shifts: Half-periods

      8        Graphing the tangent and cotangent functions

      9        An important question about sums of sinusoidal functions   

                10      Linear combinations of sines and cosines

                11      Linear combinations of sinusoidal curves with the same frequency       

                12      Linear combinations of functions with different frequencies

                13      Finding the period of a sum of sinusoidal curves

                14      A discovery of Monsieur Fourier   

  

9. Inverse Functions and Trigonometric Equations

                1        Functions and Inverse Functions

      2        Arcsin: The inverse function to sin

                3        Graphing inverse function

      4        Trigonometric equations

      5        A more general trigonometric equation

      6        More complicated trigonometric equation