Imágenes de páginas
PDF
EPUB

EXAMPLES.

1. Solve m cos (0 + x) = a, and m sin ( 4 + x ) = b, for

m sin x and m cos x.

b cosa sin cos (0-4)

Ans. m sin x =

m cos x =

[blocks in formation]

2. Solve m cos (0+x) = a, and m cos (4-x)=b, for

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small]

Multiplying (1) by cosa and (2) by sina, and adding, we get

x = m cos α + n sin α.

To find the value of y, multiply (1) by sina and (2) by cosa, and subtract the latter from the former.

y = m sin α- n cos α.

Thus

[merged small][merged small][ocr errors][ocr errors][merged small]

90. To adapt Formulæ to Logarithmic Computation. As calculations are performed principally by means of logarithms, and as we are not able by logarithms directly* to add and subtract quantities, it becomes necessary to know how to transform sums and differences into products

* Addition and Subtraction Tables are published, by means of which the logarithm of the sum or difference of two numbers may be obtained. (See Tafeln der Additions, und Subtractions, Logarithmen für sieben Stellen, von J. Zech, Berlin.)

and quotients. An expression in the form of a product or quotient is said to be adapted to logarithmic computation.

An angle, introduced into an expression in order to adapt it to logarithmic computation, is called a Subsidiary Angle. Such an angle was introduced into each of the Arts. 84, 85, 86, and 87.

The following are further examples of the use of subsidiary angles:

1. Transform a cos±b sin into a product, so as to adapt it to logarithmic computation.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

* The fundamental formulæ cos(x + y) and sin (x + y) (Art. 42) afford examples of one term equal to the sum or difference of two terms; hence we may transform an expression a cos + b sin 0 into an equivalent product, by conforming it to the formulæ just mentioned.

Thus, comparing the identity, m cos o cos 0 + m sin & sin 0 m cos (p = 0) or m cos (06), with a cos 0+ b sin 0, we will have a cos 0 + b sin 0 = m cos (04) if we assume a = m cos and b = m sin &; i.e. (Art. 83), if tan See Art. 84.

b

a

b

and m

[ocr errors]

a

cos

as above. sin o

and

.. log (a+b)= log a + 2log sec;

log (a - b) = log a + 2log cos p.

4. Transform 1239.3 sin 0-724.6 cos to a product.

log blog (-724.6) 2.86010

[blocks in formation]

=

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors]

.. 1239.3 sin 0-724.6 cos 0=1435.6 sin (0 - 30° 18'.8).

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

from which we obtain rcos p.

From (3) and (4) we obtain r and 4 (Art. 83).

(4)

[blocks in formation]
[ocr errors]

92. Trigonometric Elimination. Several simultaneous equations may be given, as in Algebra, by the combination of which certain quantities may be eliminated, and a result obtained involving the remaining quantities.

Trigonometric elimination occurs chiefly in the application of Trigonometry to the higher branches of Mathematics, as, for example, in Physical Astronomy, Mechanics, Analytic Geometry, etc. As no special rules can be given, we illustrate the process by a few examples.

EXAMPLES.

1. Eliminate from the equations

x = a cos p, y = b sin p.

From the given equations we have

[ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors]

2. Eliminate from the equations

a cos +b sin &= c,

b cos + c sin & =α.

Solving these equations for sin and cos p, we have

[blocks in formation]

gives (bc — a2)2 + (c2 — ab)2 = (ac — b2)2.

3. Eliminate

from the equations

ycosxsin : = a cos 24,

ysin

x cos

= 2a sin 2 p.

Solve for x and y, then add and subtract, and we get

x + y = a(sino + cos ) (1 + sin 2 6),

x − y = a(sin — cos 4) (1 − sin 24).

[ocr errors]

.. (x + y)2 = a2 (1 + sin 2 )3, ·

(x − y)2 = a2 (1 — sin 24)3.

·· (x+y)3+(x −

(x + y) * + (x − y) * = 2aa.

4. Eliminate a and ẞ from the equations

a = sin a cos ẞ sin 0 + cos a cos

(1)

[merged small][merged small][ocr errors][merged small][merged small]

Squaring (1) and (2), and adding, we get

a2 + b2 = sin2 a cos2 B+ cos2a.

[blocks in formation]

(3)

(4)

(5)

« AnteriorContinuar »