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Hence the following rules :
From the number of grades SUBTRACT to, and the remainder is the number of degrees.
2. To convert degrees into grades.
To the number of degrees ADD }, and the sum is the number of grades. Ex. 1. Convert 13° 18' 75" into English measure.
No. of grades = 13.1875 Subtract of this 1:31875 :. No of degrees
71.500 Ans. 11° 52' 711.5.
Ex. 2. Convert 18° 7' 30" into French measure.
No. of degrees 18.125
Add ; of this 2:0133 :: No. of grades = 20.1388
Ans. 20° 13' 88".g.
3. An angle may be conceived to be generated by the revolution of a line about a fixed point. Thus
Let OA be an initial line, and let a line, OP, starting from OA, revolve with O as centre, and take up successively the positions OP, OP, OP3, OP
Now the magnitude of an angle may be measured by the amount of turning required to generate it. When, therefore, the revolving line reaches the position OB, we may conceive an angle to have been generated whose magnitude is two right angles. And, further, when the revolving line assumes the positions
BA OP3, OP, the angles AOP3, AOP, (the letters being read in the direction of revolution) are angles whose magnitudes are each greater than two right angles. Indeed, when the revolving line again reaches the position OP, we may conceive an angle to have been generated whose magnitude is four right angles. Lastly, if the revolution of the line OP be continued, we may conceive of angles being generated to whose magnitude there is no limit.
Ex. I. 1. Express 39° 22' 30" in French measure, and 139 15' 75" in English measure.
2. One of the angles at the base of an isosceles triangle is 50°. Express the vertical angle in grades.
3. Divide an angle of n degrees into two such parts that the number of degrees in one part may be twice the number of grades in the other.
4. Two angles of a triangle are respectively a°, 69, express the other angle in degrees and grades. 5. If of a right angle be the unit of measurement, ex
을 press an angle which contains 22.5 degrees.
6. Show how to reduce English seconds to French seconds.
7. If the unit of measurement be 8°, what is the value of 109.
8. If two of the angles of a triangle be expressed in grades, and the third in degrees, they are respectively as the numbers 5, 15, 18. Find the angles.
9. What is the value in degrees and grades of an angle
which is the result of the revolution of a line 3} times round.
10. In what quadrants are the following angles found :145, 96, 327, 272, 272o.
11. If ao be taken as the unit of angular measurement, express an angle containing b. 12. What is the unit of measurement when a expresses
9 of a right angle?
THE GONIOMETRIC FUNCTIONS.
4. It was formerly usual in works on Trigonometry to give the following definitions :
Let a circle be described from centre A, I with radius AB supposed to be unity, then
(1.) The sine of an arc BC is the perpendicular from one extremity, C, of the arc upon the diameter passing through the other extremity B.
Thus ČS is the SINE of the arc BC.
B (2.) The cosine of an arc is the sine of the
S'C is the COSINE of the arc BC.
drawn from one extremity, B, of the arc touching the circle, and terminated in the diameter which passes through the other extremity, C, of the arc.
Thus, BT is the TANGENT of the arc BC.
(4.) The cotangent of an arc is the tangent of the complement of the arc.
Thus, DT' is the COTANGENT of the arc BC.
(5.) The secant of an arc BC is a line drawn from the centre through one extremity, C, of the arc, and terminated in the tangent at other extremity.
Thus, AT is the SECANT of the arc BC. (6.) The cosecant of an arc is the secant of the complement of the arc. Thus, AT' is the COSECANT of the arc BC.
(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 AD 1 (1.) Sin Ą = CS =
1 AS AT:
cosec A AS AS AB 1 (2.) Cos A = SC
1 AC AT
BT BT AD 1 (3.) Tan A = BT =
1 AB DT cot A (4.) Sin? A + cos? A = CS2 + S/C2 = CS? + AS3 = AC? = 1. (5.) Sec? A = ATP
ABP + BT2 = 1 + tan? A. (6.) Cosec? A = AT/2 = ADP + DT/2 1 + cot? A.
BT CS CS sin A (7.) Tan A = BT =
AB AS S'C (8.) Vers A = SB = AB - AS = AB S'C 1 - cos A. (9.) Covers A = S'D = AD AS'
CS = 1 - sin A. (10.) Suvers A = B'S = B'A + AS = 1 + 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 themselves, 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.
Trigonometrical Ratios. 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.
AM (4.) Cot A
perpendicular 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, or-
covers A 1 - sin A. (9.) The suversed 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.