512 trinomial

Proving the difference formula

Trisecting an angle

Alternatively, place the angle Aat the other corner of

each of the triangles containing this angle, and use the

same diagrams of four triangles arranged in a rectangle.

This yields the sum and difference formulae for sines:

sin(A+ B) = sin A cos B+ cos A sin B

sin(A– B) = sin A cos B– cos A sin B

These formulae also follow readily from E

ULER

’

S FOR

-

MULA

. We also have the following addition and sub-

traction formulae for the tangent function:

These are obtained by writing tan(A+ B) = sin(A+ B)/

cos(A+ B), for instance.

See also D

E

M

OIVRE

’

S FORMULA

;

DERIVATIVE

;

HIS

-

TORY OF TRIGONOMETRY

(essay);

HYPERBOLIC FUNC

-

TIONS

;

INVERSE TRIGONOMETRIC FUNCTIONS

; T

AYLOR

SERIES

.

trinomial Any algebraic expression consisting of

three terms, such as a+ b+ cor the quadratic expres-

sion ax2+ bx + c, is called a trinomial.

See also

MONOMIAL

;

BINOMIAL

;

POLYNOMIAL

.

triple vector product There are two basic ways to

combine the

DOT PRODUCT

and

CROSS PRODUCT

opera-

tions on

VECTOR

s to form a product of three three-

dimensional vectors a, b, and c.

The combination a· (b ×c) is called the scalar

triple product of the three vectors. The result of this

operation is a real number (scalar) that is positive if a,

b, and cform a right-handed system and is negative if,

instead, they form a left-handed system. Changing the

order of the vectors in this triple product can change

the orientation of the system involved. We have:

a· (b ×c) = b· (c ×a) = c· (a ×b) = –b· (a ×c)

= –a· (c ×b) = –c· (b ×a)

The scalar triple product has a nice geometric inter-

pretation:

The absolute value of the scalar triple product

equals the volume of the

PARALLELEPIPED

formed by the vectors a, b, and c.

To see this, note that b×cis a vector perpendicular to

the base of the parallelepiped formed by the two vec-

tors band c. If θis the angle between this vector and a,

then the height of the parallelepiped is the absolute

value of |a| cos(θ). Since, according to the cross prod-

uct, the area of the base is given by | b×c| we have that

the volume of the parallelepiped is: base ×height =

|a|| b×c|| cos(θ)|, which is precisely the formula for the

absolute value of the dot product: a· (b ×c).

An exercise in algebra shows that the scalar triple

product of three vectors can be computed by taking the

DETERMINANT

of the 3×3

MATRIX

whose rows are the

entries of the vectors a, b, and c, respectively.

The vector triple product of three vectors is defined

to be the combination: a×(b ×c). The result is a vector

that lies perpendicular to b×c, as well as to a. As b×c

is perpendicular to the plane formed by band c, it fol-

lows that a×(b ×c) lies in the bc-plane.

See also

RIGHT

-

HANDED

/

LEFT

-

HANDED SYSTEM

.

trisecting an angle One of the problems of antiquity

(like

DUPLICATING THE CUBE

and

SQUARING THE CIRCLE

)

of considerable interest to the classical Greek scholars