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By similar triangles, the functions are constants for a constant angle, but variables for a variable angle.

The base line is the initial line; the hypotenuse line, the terminus of the angle.

Linear Representation. (1) If h=1, o= sin, a=cos. (2) If a = 1, o = tan.

The transverse line is TT' through vertex and perpendicular to initial line II'.

sin = transverse projection of directed unit. (Unit h.) COS = initial projection of directed unit.*

=

tan transverse projection of h if initial projection be unity.† Since antecedent = consequent × ratio, also

For sine and cosine, consequent

h, ratio

function.

RULE I. To obtain either side from h, multiply by ratio, sine for o, cosine for a.

RULE II. To obtain the sine from cosine, multiply by tangent.

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Quadrants. II' and TT' divide the angular space about the vertex into four quadrants, numbered as in the figure. An angle is in the quadrant in which it terminates.

* The angle being the direction of its terminus, we may speak of the ratios as direction ratios.

Since for the other acute angle of ratio triangle,

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If a circle be described with the unit base a as radius, o is a tangent.

The Terminal Values of the functions are as follows:

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The algebraic signs being determined thus: to right and up, +; to left and down,

PRACTICAL DEVELOPMENT.

Wishing to calculate the distance IB to an object B, starting from I, I laid off IA IB.

At a distance AM-1 from AI erected MN LAI, determining N by looking from A to B.

I also measured AP, and drew PL 1 AB.

1

1

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18

B

8

с

с

с

unit

cos.

unit

L M

The last is not needed in measuring the distance; in fact, AM might have been any distance, when IB could have been MN × AI found, as IB

=

AM

The advantage of a table of tangents is, that we never have need to construct the small triangle.

=

If IA 1000 feet, and we have the tangent from a table, we have simply to move the decimal point three places, and we have IB at once.

Two-Place Table. Take 10 inches as an hypotenuse, and, by aid of a protractor (or by constructing an angle of 30°, geometrically, and then trisecting it by folding), construct the

values of sine and cosine ... tan = for every 10°. Here

10 inches =

sin

COS

unit. ... 0.1 inch = 0.01 unit.

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160

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170

180

-10°

Evidently (arithmetically) function (180° — A) = ƒ (4). The ratio triangles being equal, having h and A equal.

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1. Give functions, if o, a, h, are (1) 6, 8, 10; (2) 10, 24, 26; (3) 4, 7, 5, 8.5.

100;

2. Solve the following: Z, o, a, h being (1) 20°, ?, ?, (2) ?, 4, ?, 5; (3) 57', 4000, ?, ?; (4) 8.8′′, 4000, ?, ?.

NOTE. If the greatest angular distance of Venus from the sun be 45°, what is its distance from that body as compared to that of the earth?

3. Can the sines of 0°, 30°, 45°, 60°, and 90° be written, √ √ √2, {√3. ±√4?

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4. If A, B, and C be the angles, and a, b, and c the opposite sides of a triangle, p the perpendicular from C to c, show that a sin B = P: = b sin A. ... ab sin A: sin B. (In words.)

Do field work, using ratios to two places,

sin 15 (0.17 +0.34),

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NOTE. If the greatest and least values of the maximum elongation of Mercury be 15° and 30°, what are its greatest and least distances from the sun?

Logarithmic Solutions. Though strictly Algebra, we give the logarithmic solutions thus far:

log of sin = logo-log h... logo= log h+ log of sin.

log of cos = log a log of tan = log o log sin

log sin A

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log h... log a = log h+ log of cos.

log a... logo= log a + log of tan. = log of sin + 10, a: b = sin A: sin B, gives = log a + colog b + log sin B.

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a

a = b tan A,

a

b

=

tan A

(3) By T2, tan4=;

(1) is the definition of the sine ratio.
(2) is the definition of the cosine ratio.

(3) is the definition of the tangent ratio.
By T, sin2 + cos2 = 1.

... a2 = h2 sin2, and b2 = h2 cos2.

... (4), h2 = a2+b2; (4), sin2 + cos2 = 1 = cot A cot B.

(4), is the Pythagorean formula.

(5) By T4, sin A= cos B, cos A = sin B.
(5) is the complementary formula.

NOTE. For construction of figure, p. 13, see p. 15.

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