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m cOS X =
m cOS X =
1. Solve m cos (8 + x) = a, and m sin ( 0 + x) = b, for m sin x and m cos x.
Ans. m sin x=
b cos 0 — a sin •
cos (0 - 0) b sin + a cos
cos (0 - 0) 2. Solvem cos (0 + x)= a, and m cos ($ — x)=b, for m sin x and m cos x.
b cos A. Ans. m sin x =
a cos sin (@+) b sin + a sin.
sin (@++) 89. Solve the equation æ cos a + y sin arm
(1) æ sin a Y COS (4=n
(2) for x and y.
Multiplying (1) by cosa and (2) by sina, and adding, we get
X = m cos ok + n sin a. To find the value of y, multiply (1) by sin a and (2) by COS Q, and subtract the latter from the former. Thus
y = m sin c
n COS O.
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 Addi. tions, 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 0 + b sin A into a product, so as to adapt it to logarithmic computation.
6 Put = tan $ ;
COS 2. Similarly, a sin 0 + bcos A = sin (++).
cos o 3. Transform a + b into a product,
+b=a1+) =a (1+tanod) = a sec
* 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 0 + b sin 8 into an equivalent product, by conforming it to the formulæ just mentioned.
Thus, comparing the identity, m cos o cos 0 + m sin o sin 0 = m cos (° F ) or m cos (0 7 °), with a cos 0 + b sin 0, we will have a cos 0 + b sin = m Cos (0 F if we assume
b a = m cos and b = m sin $; i.e. (Art. 83), if tan =
cos o See Art. 84.
.. log (a + b)= log a + 2log sec $;
log (a - b)= log a + 2 log cos $.
4. Transform 1239.3 sin 0 724.6 cos 0 to a product.
logb= log ( – 724.6)= 2.86010-
log tan ø= 9.76692
.. 1239.3 sin 0 — 724.6 cos 0 = 1435.6 sin (0 – 30° 18'.8).
From (1) and (2) we have
b rcos o =
sin from which we obtain a cos 0.
From (3) and (4) we obtain r and $ (Art. 83).
1. Solve rcos o cos 0 =- 53.953,
r cos o sin 0 =
rsino = - 39.062, for r, 0, 0. log b= 2.29493
..$=-10° 49'. log a = 1.73201
log r cos o = 2.31060 log tan 0 = 0.56292- log cos d=9.99221 .:: A=105° 18'.0.
logr = 2.31839
.. 1= 208.16.
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 ¢, y=b sin $. From the given equations we have
b which in
cos$ + sin?
2. Eliminate from the equations
a cos 0 + b sin ø=c,
b cos + csino
bc - a2
which in cos? $ + sin’d=1,
3. Eliminate from the equations y cos $ — xsin =
a cos 2$, ysin $ + xcos 2 a sin 20. Solve for x and y, then add and subtract, and we get
x+y=a(sin $ + cos 0) (1 + sin 2 ),
2 — Y= a(sin - cos 0) (1 - sin 2). .: (x + y)2 = a’(1 + sin 20), :
(2x — y)2 = a’(1 – sin 20). .: (x + y)+ (x – y) i = 2 a3.
4. Eliminate a and ß from the equations
a = sin a cos ß sin 0 + cos a cos 0
c=sin a sin ß sino .
a’ + b2 = sino cos? ß + cos’o.
sino a sin'