Ex. 1. To find how many degrees and minutes are contained in the angle 42o, 34', 56". retaining the tenths and hundredths and neglecting the thousandth parts of a second. Ex. 2. Find how many grades and minutes are contained in the angle 24o, 51', 45". First reducing the minutes and seconds to the decimal parts of a degree, 11. DEF. The complement of an angle is its defect from a right angle. Thus 90° - 24o, 32′ = 65°, 28′ is the complement of 24o, 32'. 90° - 110°, 15' = (20°, 15') is the complement of 110o, 15'. 12. DEF. The supplement of an angle is its defect from two right angles. Thus 180° 56°, 20′ = 123°, 40′ is the supplement of 56o, 20'. - (6o, 12′) is the supplement of 186o, 12′. 180° - 186o, 12' = ― A few examples to each chapter are placed after the third appendix. CHAPTER II. OF GONIOMETRICAL FUNCTIONS OF ONE ANGLE, AND SOME FORMULÆ CONNECTING THEM WITH EACH OTHER. 13. DEF. PLANE TRIGONOMETRY, in its original meaning, implies the measuring of plane triangles ;-in its extended signification, it treats of the formulæ connecting the relations of angles with each other, and of the solution of plane rectilineal figures. 14. Let a straight line revolve from the fixed line AB round the point A, in the direction of the letters B, D, B', D', and let there be A degrees in the angle BAC. From any point C in AC draw CN at right angles to AB,. produced if necessary, and through A draw DAD' perpendicular to AB. Now, for the reasons given in Arts. 2 and 3, in these figures the signs of NC are +, +, those of AN are 15. DEFINITIONS. respectively, and + respectively. See the figures of the last Article. NC NC 1. is the sine of the 4 BAC; or, sin ▲ BAC= = AC AC AN AN 2. is the cosine of the BAC; or, cos 4 BAC = AC AC NC NC 3. is the tangent of the BAC; or, tan ▲ BAC = AN AN AC AC 4. is the secant of the 4 BAC; or, sec ▲ BAC = AN AN 5. 1 - cos ▲ BAC is the versed sine of the ▲ BAC; or, versin BAC=1 cos / BAC; 6. The tangent of the complement of the BAC is called the cotangent of the ▲ BAC; or, cot BAC = tan (90° COR. Since, if 90o BAC be the original angle, its complement is ▲ BAC, Art. 11; cotan (90° or, tan we have by this definition, BAC) tan ▲ BAC, = BAC cotan (90° - BAC). = ▲ 7. The secant of the complement of the BAC is called the cosecant of the BAC; COR. Since if 90o BAC be the original angle, its complement is ▲ BAC, Art. 11, we have by this definition, 8. cosec (90° - ▲ BAC) = sec ▲ BAC; or, sec ▲ BAC = cosec (90° L BAC). The versed sine of the supplement of the BAC is called the suversine of the BAC; 16. or, suversin BAC versin (180° - 4 BAC). The cosine of 4 BAC might have been defined to be the sine of the complement of ▲ BAC. or the sine of an angle is equal to the cosine of its complement. In the following pages we shall for the sake of convenience indicate an angle by a single letter, as sin A, cos A, tan B. B 17. Sin A, cos A, tan A...are proper measures to determine the magnitude of an angle, since so long as the angle A remains the same, they are invariable whatever be the magnitude of AC. Let D be any other point in AC, DM perpendicular to AB. wherever in the line AC the point C be situated;—similarly it may be shewn that cos 4, tan A, sec A...are invariable quantities so long as the magnitude of A remains unaltered. Hence, if any of the quantities sin A, cos A, tan A, sec A... be given, the angle A may be determined. Ex. 1. Required to determine the angle whose sine is 2 3 Take any line AB and describe upon it a semicircle. From AB cut off the part BE equal to one third of AB, (Euclid vI. 9.), and with center A and radius AE describe a circle cutting ACB in C; join AC and CB. Then the angle ACB, being in a semicircle, is a right angle, and 4 Ex. 2. Required the angle whose tangent is. 5 Let a be any line and take AN = 5a, (fig. 1. Art. 20.) E Draw NC at right angles to AN, and let it = 4a: join AC. |