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84 THE FASCINATION OF NUMBERS
Sl Nopraiiala
Hypotenuse
B

Tl et S

Hypotenuse
Tangent 4= ? @fpendigqlar

Base

If the angle C remains a right-angle, it follows that, as the
angles of a triangle always total 180°, the angles B and A4
must always add up to 9o°. Thus, as 4 is increased by 1°, so
the angle B is decreased by 1°, and in short the angles 4 and
B are always complementary. It follows from this that the
cosine of angle 4 is the same as the sine of angle B, being
Side AC
Side 4B’
relation to the angle A4, but is the perpendicular side in
relation to the angle B.

Since there is such a close relationship between sines and
cosines, it is not surprising to find that they can be related
in many different ways. In particular, there are two expres-
sions with which we are concerned here (proofs being avail-
able in any book on trigonometry). The two expressions are:

(1) Sin(4+B)=Sin A.Cos B-Sin B.Cos A.

(1) Sin(4—B)=Sin A.Cos B—Sin B.Cos 4.
If these two expressions are added together, we obtain the
result:

(iii) Sin(4+B)+Sin(4—B)=2!4.Cos B
so that

(iv) Sin 4.Cos B=4}[Sin(4 +B) +Sin(4—B)]

Therefore, if it is required to multiply together two num-
bers which are the equivalents of Sin 4 and Cos B respec-
tively, this can be achieved by calculating the value of the
expression $[Sin(4 +B) +Sin(4 —B)]. For example, in order
to multiply 2588 by -9848, we proceed as follows:

2588 =Sin 15°
9848 =Cos 10°

equivalent to for the side AC is the base side in

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