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39. 41, 1T, 2.504. Ans. 257° 49' 43".39, 15°, 1430.468.
. 2 40. .0234, 1.234, 1°20'27", 70° 42' 11", 38° 11'50".
3 41. Find the number of radians in an angle at the centre of a circle of radius 25 feet, which intercepts an arc of
Ans. 11. 42. Find the number of degrees in an angle at the centre of a circle of radius 10 feet, which intercepts an arc of 57 feet.
Ans. 90°. 43. Find the number of right angles in an angle at the centre of a circle of radius 34 inches, which intercepts an arc of 2 feet.
44. Find the length of the arc subtending an angle of 4į radians at the centre of a circle whose radius is 25 feet.
Ans. 112ft. 45. Find the length of an arc of $0° on a circle of 4 feet radius.
Ans. 533 ft. 46. The angle subtended by the diameter of the Sun at the eye of an observer is 32' : find approximately the diameter of the Sun if its distance from the observer be 90 000 000 miles.
Ans. 838 000 miles. 47. A railway train is travelling on a curve of half a mile radius at the rate of 20 miles an hour: through what angle has it turned in 10 seconds ?
Ans. 6.4 degrees. 48. If the radius of a circle be 4000 miles, find the length of an arc which subtends an angle of 1" at the centre of the circle.
Ans. About 34 yards. 49. On a circle of 80 feet radius it was found that an angle of 22° 30' at the centre was subtended by an 31 ft. 5 in. in length: hence calculate to four decimal places the numerical value of the ratio of the circumference of a circle to its diameter.
Ans. 3.1416. 50. Find the number of radians in 10" correct to four significant figures (use tii for 7).
THE TRIGONOMETRIO FUNCTIONS.
13. Definitions of the Trigonometric Functions. — Let RAD be an angle; in AD, one of the lines containing the angle, take any point B, and from B draw BC perpendicular to the other
B line AR, thus forming a right triangle ABC, right-angled at C. Then denoting the angles by the capital letters A,
-R B, C, respectively, and the three sides A
с opposite these angles by the corresponding small italics, a, b, c,* we have the following definitions : opposite side
is called the sine of the angle A.
is called the cosine of the angle A.
is called the tangent of the angle A.
is called the cotangent of the angle A. opposite side
hypotenuse is called the secant of the angle A. 7
is called the cosecant of the angle A. opposite side If the cosine of A be subtracted from unity, the remainder is called the versed sine of A. If the sine of A be sub
* The letters a, b, c are numbers, being the number of times the lengths of the sides contain some choson unit of length.
tracted from unity, the remainder is called the coversed sine of A; the latter term is hardly ever used in practice.
The words sine, cosine, etc., are abbreviated, and the functions of an angle A are written thus : sin A, cos A, tan A, cot A, sec A, cosec A, vers A, covers A.
The following is the verbal enunciation of these definitions :
The sine of an angle is the ratio of the opposite side to the hypotenuse; or sin A = a.
The cosine of an angle is the ratio of the adjacent side to the hypotenuse; or cos A
The tangent of an angle is the ratio of the opposite side to the adjacent side; or tan A
b The cotangent of an angle is the ratio of the adjacent side to
b the opposite side; or cot A
The secant of an angle is the ratio of the hypotenuse to the
с adjacent side; or sec A :
b The cosecant of an angle is the ratio of the hypotenuse to the
с opposite side; or cosec A
The versed sine of an angle is unity minus the cosine of the
b angle; or vers A 1 - COS A: 1
The coversed sine of an angle is unity minus the sine of the angle; or covers A=1-sin A=1
These ratios are called Trigonometric Functions. The student should carefully commit them to memory, as upon them is founded the whole theory of Trigonometry.
These functions are, it will be observed, not lengths, but ratios of one length to another; that is, they are abstract numbers, simply numerical quantities; and they remain unchanged so long as the angle remains unchanged, as will be proved in Art. 14.
It is clear from the above definitions that
The powers of the Trigonometric functions are expressed as follows:
(sin A) is written sin’ A,
(cos A)' is written cos: A, and so on.
NOTE. — The student must notice that 'sin A'is a single symbol, the name of a number, or fraction belonging to the angle A. Also sin? A is an abbreviation for (sin A), i.e., for (sin A)*(sin A). Such abbreviations are used for convenience.
14. The Trigonometric Functions are always the Same for the Same Angle. — Let BAD be any angle; in AD take P, P', any two points, and draw PC, P'C' perpendicular to AB. Take P", any point in AB, and draw P"C" per
B pendicular to AD.
C C P" Then the three triangles PAC, P'AC', P"AC" are equiangular, since they are right-angled, and have a common angle at A: therefore they are similar.
PC P'C' P'C"
But each of these ratios is the sine of the angle A. Thus, sin A is the same whatever be the position of the point P on either of the lines containing the angle A.
Therefore sin A is always the same.
A similar proof may be given for each of the other functions. In the right triangle of Art. 13, show that a= c sin A = cos B
b tan A=b cot B,
= a tan B
=c sin B, c = a cosec A = a sec B=b sec A=b cosec B. NOTE. - These results should be carefully noticed, as they are of frequent use in the solution of right triangles and elsewhere.
1. Calculate the value of the functions, sine, cosine, etc., of the angle A in the right triangles whose sides a, b, c are respectively (1) 8, 15, 17; (2) 40, 9, 41; (3) 196, 315, 371; (4) 480, 31, 481; (5) 1700, 945, 1945. Ans. (1) sin A = ', cos A = 14, tanA= 15, etc.;
. (5) sin A = 348, tan A = 349, etc. )
A In a right triangle, given: 2. a= m+na, b= V2 mn; calculate sin A.
Vm2 + n? Ans.
3. a=Vm— mn, b=n; calculate sec A.
4. d=Vm2 + mn, c=m+n; calculate tan A.
m2 + mn
5. a= 2mn, b= m2 — no; calculate cos A.