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2. In latitude 48° 50' north, the true altitude of the sun's centre was 22° 2' the declination at the time was 100 12' S., and its magnetic bearing 161° 32' east. Required the variation.

o's polar distance 1000 12 sin. zenith distance 67 58 ar. comp. 0.032936

sin. colatitude 41 10 ar. comp. 0 181608

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@'s true azimuth N. 138 51 20 E. observed azimuth N. 161 32 O E.

22 40 40 West.

The variation is west, because the sun's observed distance from the north, measured easterly, being greater than its true distance, intimates that the north point of the compass has approached towards the west.

3. In latitude 48° 20' north, the star Rigel was observed to set 90 50 to the northward of the west point of the compass; required the variation, the declination of Rigel being 8° 25' S.

Variation, 22° 33' West.



115. We have postponed to this place the investigation of a few formulæ requisite for the study of Analytical Geometry. By resuming the expression for the tangent, (Art. 32,) and putting a + b for a, we have

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tan (a + b)

R (sin a cos b + sin b cos a) = cos a cos b sin a sin b

dividing both numerator and denominator of the second member of this equation by cos a cos b, we have

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i. e. the tangent of the sum or difference of two arcs is equal to rad. square into the sum or difference of their tangents divided by rad. square minus or plus the rectangle of their tangents.

If a represent the tangent of a and a' the tangent of a', then R being 1,



tán (α-a') Ligns


Using the upper sines and making ba in equation (1),

we have

tan 2a =

2 R2 tan a
R2-tan2 a


Tan 3a, tan 4a, &c., may be found by making b suecessively equal to 2a, 3a, &c.

116. The sine and cosine of 45° are equal, since the complement of 45° is 45°. These two lines form two sides of a square of which R is the diagonal. But (Geom., B. 4, Prop. 11, Cor. 4,) the diagonal is to the side of a square as √2 to 1, hence

sin 45°

cos 45°

R √2

Multiplying both numerator and denominator by v2, this value may be changed into

sin 45°

& R V2

117. The sine of an arc is equal to the chòrd of the


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which passes through the other extremity A. Corollary. The chord of 60°, or




of the circumference which is the side of the regular hexagon, is equal to x (Geom., B. 5, Prop. 4,); hence the sine of 30° is equal to R. 118. Referring to Art. 33, it will be observed that

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119. Making a = 45° in equation (1) of Art. 115, we have, R being 1,

tan (45° + b) =

1 + tan b

1 tan b

120. Our demonstration for the sine and cosine of the sum of two arcs at Art. 69, might seem to want generality, since the arcs a and b are there supposed to be less than 90°. That these arcs may extend to the other quadrants, can be shown as follows:

Let a 90°+m, then will the formula

sin (a+b)= sin a cos b+-sin b cos a



* Tan 45° cot 45o. R.

still be true, for substituting 90°+-m for a, we have sin (90° +m+b) in place of the first member, which is equal to cos (m+b)*; for the second member by the same substitution, we have

sin (90°+m) cos b+sin b cos (90+)


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but sin (90°+m)= cos m and cos (90° +m)=--sin m,

+= hence equation (1) becomes

cos (m+b)=cos m cos b sin m sin b which, since m and b are less than 90°, we know to be true, by Art. 69; hence (1), from which it is derived, is true also.

Assuming (1) to be true with a > 90°, which we have just proved, make b=90°+m, and in a similar manner the truth of the formula may be established on the supposition of both a and b> 90°.

Afterwards make a = 180o+m and observe that sin (180° mb)= +m+6)=sin m+b and cos (180°+m)=-- cos m, and you will show that the formula extends to the third quadrant, and so on.

1.21, As we were not sufficiently advanced in the theory of trigonometrical lines to explain the construction of the tables of sines, tangents, &c., at Art. 38, having once been obliged to use them without explanation, we thought best to defer that to this place, so as not to interrupt, more than was necessary, the train of reasoning relative to the solution of triangles.

The diameter of a circle being multiplied by =3.1415926 we have the length of its circumference; this divided by 360 gives us the length of one degree, and this by 60 the length of one minute of the circumference. So small an are as l' may be considered as equal to its sine, without sensible

Having thus found the sine of l' we may find the sines of other arcs by formula (1), of Art. 74.

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* By referring to either of the diagrams in which a sine is drawn, it will be ovident that sin (90°+a), a being any arc less than a quadrant, is equal in length to sin (90°—a) = cos a. Also that cos (90°+a) = — cos (90°—a)=

- sin a,

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