trigonometrical ratio is a tangent, cotangent, secant or cosine, by finding in the first place, whether its algebraic sign is + or In the former case the angle taken out of the tables is the one sought in the latter we must subtract the angle taken out from 180° for the required angle.

:

34. The rule for finding the algebraic sign of a trigonometrical ratio in any equation or formula where all the other terms are known is the following.

Having written down the formula and simplified it by clearing it of fractions, &c., put over each given term its proper algebraical sign; that is, over each tangent, secant, cosine and cotangent of an angle less than 90° the sign +, and over each of the same quantities when the angle is greater than 90° the sign : but over each sine, cosecant and versine the sign + whether the angle is greater or less than 90°, and determine from thence the sign of the product of that side whose terms are all known.

Then, since the sign of the product of each side of the equation must be the same (otherwise we should have a positive quantity equal to a negative quantity): make it so by putting over the unknown term the sign + accordingly.

or

If + falls over the unknown term, the part required is less than 90°, and the quantity taken out will be the angle; but if subtract the angle taken out of the tables from 180°, the remainder will then be the angle required. When the part sought is expressed in terms of the sine, the above rule will not apply since the sine is

positive, both when the angle is greater or less than 90o. The uncertainty which thence arises can only be removed in particular cases.

In the above rule the angle required is supposed to be always less than 180°.

EXAMPLES.

In the following examples it is required to find whether the angle r is greater or less than 90°: supposing A=45°, B=120° and C-130°.

(123) Cos. r = tan. A cos. B sec. C.

Placing over the given terms the proper signs we have

cos. x = tan. A cos. B sec. C, and since the product of the three terms on the right hand side of the equation is positive, cos. x must also be positive, and therefore x is less than 90°.

*By turning the formula into logarithms, we shall see that 10 must be subtracted from the left hand side, and 30 from the right; or the left hand side of the equation must be diminished by 20: thus

log.cos.x-10=log.tan. A-10+log.cos. B-10+log.sec. C – 10, or log. cos. x=log. tan. A+ log. cos. Blog. sec. C-20.

In the same manner the rejection or addition of the tens in any logarithmic formula may be explained.

(124) sec. x sin. A sec. B: determining by the rule the signs of sin. A and sec. B, we find that sec. x

+

= sin. A, sec. B, a negative product, therefore sec. x is negative, or x is greater than 90°.

(125) Tan. x

+

.*.x=135

cosec A cos. B placing over A and B their proper signs we find that tan. x is negative,

(127) sec. A sin.2C=cos.2B cot. x

.. is less than 90°.

(128) sec. A sin.2C= cos 2 B cot. r: placing the signs and taking into consideration the negative sign in front of the right hand side of the equation, we have,

+

sec. A sin2C = — cos.2B cot. x

+

..r is greater than 90°.

(129) Sin. A cos. x = cos. B cot. C

(130) Ver. A =

+

r is greater than 90°.

cos, B

sin. C tan. r

+

or ver. A sin. C tan. x = cos. B

or r is greater than 90°.

RULES

IN PLANE AND SPHERICAL TRIGONOMETRY.

I.

Three sides of a plane triangle being given, to find an angle.

Put down the two sides containing the required angle, and take the difference: under which put the third side: take the sum and difference, and also the half sum and half difference.

To the arithmetical complements* of the logarithms of the two first terms in this form, add the logarithms of the two last, and reject 10 in the index: the result will be the log. haversine (a) of the required angle, which find in the table.

(a) If the student have no table of haversines the angle may be found by the following rule.

10-log. b is called the arithmetical complement of log. b. In practice it is easily found by taking each figure of the logarithm from 9, except the last, and that from 10: thus, ar. co. log. of 2.714152 is 7.285848; ar. co. log. 1.314150 is 10.685850.

II.

Three sides of a plane triangle being given, to find an angle.

Put down the two sides containing the required angle, and take the difference: under which put the third side: take the sum and difference, and also the half sum and half difference.

To the arithmetical complements of the logarithms of the two first terms in this form, add the logarithms of the two last, divide by 2, and look out the result as a log. sine; the angle corresponding to which will be half the required angle.

EXAMPLES.