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(7.) The versed sine is that portion of the radius upon which the sine falls, which is included between the sine and the extremity of
Thus, SB is the VERSED SINE of the arc BC.
(8.) The coversed sine is the versed sine of the complement of the arc.
Thus, S'D is the COVERSED SINE of the arc BC. (9.) The suversed sine is the versed sine of the supplement of the arc. Thus, B'S is the SUVERSED SINE of the arc BC.
Representing the arc BC by A, it is usual to write the above functions thus :-Sin A, cos A, tan A, cot A, sec A, cosec A, vers A, covers A,. suvers A.
By mere inspection, the student will see that the following relations hold :
CS CS AD 1 (1.) Sin A = CS =
1 AS AT
AS AS AB 1 (2.) Cos A = S'C =
1 AC AT
AD 1 (3.) Tan A = BT
cot A' (4.) Sin? A + cos? A = CS? + S/C? = CS? + AS? = AC? = 1. (5.) Sec? A = AT? = AB + BT? = l + tan? A. (6.) Cosec? A = AT/2 = AD? + DT'2 1 + cot? A.
BT CS CS sin A (7.) Tan A
AB AS S'C (8.) Vers A = SB = AB AS = AB S'C = l - cos A. (9.) Covers A = S'D = AD AS' = AD CS = 1 sin A. (10.) Suvers A B'S = B'A + AS = l + cos A.
It is more convenient, however, to define the sine, cosine, &c., as in the next article, according to which definitions they are commonly called TRIGONOMETRICAL RATIOS. The student will see that if the above definitions be so far modified that, instead of the lines them. selves, the goniometric functions be taken as the ratios which the lines respectively bear to the radius, they are included in the definitions of the next article.
5. Let BAC be any angle, which we may denote by A, and P any point in the line AC. Draw PM perpendicular to AB.
Then (1.) Sin A
perpendicular PM (3.) Tan A =
AM (4.) Cot A
PM (7.) The versed sine is the remainder after subtracting the cosine from unity, orvers A
1 - cos A. (8.) The coversed sine is the remainder after subtracting the sine from unity, orcovers A
1 · sin A. (9.) The suvorsed sine is the sum of the cosine and unity, or
suvers A 1 + cos A. The last three are not much used in practice.
6. Comparing (1.) and (6.), (2.) and (5.), (3.) and (4.), of the last article, we see at once that the sine and cosecant, the cosine and secant, and the tangent and cotangent, are respectively each the reciprocal of the other.
Hence also, transposing and taking the square root-
AP2 AM? + PM? PM? (7.) Sec A =
1+ 1 + tan’ A. AM? AM?
AM? (8.) Cosec? A =
PM = 1 + cot? A.
PM PM AM (9.) Tan A
= sin A ; cos A AM AP AP sin A
= cos A ; sin A AP
(10.) Cot A =
The student must make himself thoroughly master of the results in this article.
7. To express the trigonometrical ratios in terms of the sine. (1.) Cos A
1 – sino A, by Art. 6 (6.)
sin A (2.) Tan A
by Art. 6 (9.),
Ex. If sin A , find the other trigonometrical ratios. We have, cos A = 1
= V1 - 5 = 4.
cOS A 8. To express the trigonometrical ratios in terms of the cotangent.
1 (1.) Sin A =
by Art. 6 (8.) cosec A
v1 + cot A 1
by Art. 6 (7.)
1 (3.) Tan A
by (2.) above, Ncot? A + 1
cot A (5.) Cosec A
Vi + cot A, by Art. 6 (8.) And in the same way the trigonometrical ratios may be expressed in terms of any one of them.
Ex. II. 1. Given sin A 1}, find the other trigonometrical ratios.
2. Given tan A = 13, find the remaining trigonometrical ratios.
1 3. If cot A = a, show that sin A
Vi + a
126 4. If vers A 6, then tan A
6 5. Construct by scale and compass an angle (1.) whose cosine is 4; (2.) whose tangent is g; (3.) whose secant is 2; (4.) whose cotangent is 2 + 13.
Prove the following identities :6. (Sin A + cos A) + (sin A cos A) 2. 7. Sec? A + cos2 B. cosec B = cosec?B + sin? A. sec A. 8. Sec? A. cosec? A = sec? A + coseco A. 9. Sec A. A tan A + cot A. 10. Sin8 A cos: A
- cos A) (sin A + cos* A). Sec A + tan A 11.
cosec B cot B Cosec B + cot B
sin“ A t sin? A cos? A + cos" A 12. 1 + sin A cos A
1 - sin A cos A 13. (0 cos 0 + y sin o) (2 sino + y cos 0) – (a cos 0 - y sin o) (ac sin 0-y cose) = 2 xy.