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cosec. 2 A=

sec. A cosec. A.

2 sin. A cos. A

(31.) Another useful class of formulas are those for half arcs; they may be easily deduced from the expressions for the double arcs: thus putting § A for A, we have from (20).

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Other useful values of sin. A, and cos. A, are derivable from the equation (18) last article, for when A is put for A the equation is

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and if this be either added to or subtracted from

1=sin.2 A+ cos.2 A,

the second member will become in each case a perfect square, viz.

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Let A be less than 90°, then the radical must be taken positive in the first, and negative in the second expression; hence, by additio and subtraction,

sin. A=} {√T+ sin. A Isin. A}

... •

(V).

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By means of these two expressions the accuracy of a table of sines

and cosines may be examined; that is to say, from the calculated values sin. A, in the table, we may compute, by these equations, the values of sin. § A, and of cos. } A; if these agree with the tabular values, found by other means, we may conclude that the tables are correct in the parts thus verified. Formulas employed in this manner to put the accuracy of the tables to the test are called formulas of verification. We have given three of these, and marked them with the letter (V).

(32.) The following formulas involving the half sums and half differences of two arcs are of frequent application: substitute § (A + B) for A, and § (A — B) for B, in the equations (3), (4), at art. (26) and we have

sin. A + sin. B= 2 sin. } (A + B) cos. } (A — B) cos. A + cos. B=2 cos. ¦ (A + B) cos. § (A — B) sin. A—sin. B=2 cos. } (A + B) sin. § (A — B) cos. B — cos. A=2 sin. } (A + B) sin. § (A —-B) and from these we get, by division,

(27);

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In each of these expressions let A= 90°, and we shall have 1+sin. B=2 sin. (45° + B) cos. (45° — } B)=2 sin.2 (45°+¦B)

cos.B=2 cos. (45° + 1B) cos. (45° — }B)=2 cos.21B — 1, by eq. 18, 1-sin. B=2 cos. (45° +}B) sin. (45° — } B)=2 cos.2 (45° +¦ B) =2 sin.2 (45° — {B)

1—cos.B=2 sin. (45° + {B) sin. (45° — § B)= 2 sin.2B, by eq. 19,

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Lastly, substituting A + B for A in (26) last article, we have

sin. (A+B)=2 sin. 1 (A + B) cos. § (A + B);

and dividing this by each of the formulas (27) in succession, there results

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(33.) We shall conclude this chapter on the theory of the trigonometrical lines, with two curious and useful propositions.

1. To express the sine and cosine of a real arc by means of imaginary exponentials.

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where e represents the base of the Naperian logarithms, that is, e= 2.7182818, &c.

√1 successively, and we have

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See the "Elementary Essay on the construction of Logarithms,"

page 68.

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But by art. (30) the series on the right is the development of cos. x,

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But by art. (30) the series on the right is the development of sin. r; hence

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2. To develop sin." x, cos." x, in terms of the sine and cosine of the multiples of x.

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from which, by addition and multiplication, we get

un+vn=2 cos. nx, un v2 = 1

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(3).

Add together the equations (1); there will result

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cos."x=

adding these equations together, and dividing by 2, we have

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hence, by substitution, the development of cos."r becomes

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Again, subtract the second of (1) from the first, and we have

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