COS A ; put A 153 cos 30° cos 15° (3.) Ratios of 15°. A 1 By last Art., tan 30°, or 2 sin A 13 1 1 2 then 'tan 15° = 2 - 13. sin 30° 1 2 From this result we easily get, Art. 8, 13 - 1 13 + 1 Sin 15° &c. 2 2 2 V2 (4.) Ratios of 75°. 13 + 1 We have, sin 75o = sin (90° – 15°) 2/2' cos 75o = sin (90° – 15°) 13 - 1 = sin 15° 2 2 1 tan 75o = tan (90° 15°) = cot 150 2 13 = 2 + 13, &c. (5.) Ratios of 135o. 1 We have, sin 135o = sin(180° - 135) = sin 45o = 12, 12 cos 135o = cos (180° – 1359) 1 = cos 15° COS 45° tan 135° tan 45o tan (180° - 1359 Ex. III. 1. Define a negative angle, and show that tan (-A) tan A, when A lies between – 90° and - 180°. 2. Trace the changes of sign of sin A.cos A through the four quadrants. 3. Trace the changes of sign of cos A + sin A, and of cos A - sin A, as A changes from 45° to 315o, 4. Assuming generally that cos 2 A cos? A sino A, trace the changes of sign of cos 2 A as A changes from – 45° to 315o 5. Write down the sines of 210', 165°, 240°, - 120°. 6. Show that sin (90° + A). = cos A, and cos (90° + A) - sin A, for any value of A from 0° to 180°. А. 7. Assuming generally that 2 cosa 1 + cos A, and 2 A Α. 2 sin? 1 cos A, show that 12 cos 71 + cos A 2 2 A and 12 sin - cos A, when A lies between 360° 2 and 540°. A 8. Given cos A = 1 2 sino show that sin A 2' Α. A = 2 sin 2 2 А. 9. Hence show that 2 cos VI + sin A A 2 COS Solve the following equations :10. CosA +cos A Ta 11. Tan o + 5 coto 6. 2 12. Sin A + sec A 13. 2 cos A 3 sin A. 13 14. Sin (A + B) = cos (A - B) 2 15. TanA = 2 sino A. 16. Sin (3 A + 75) = cos (2 A - 15°). 5 17. Sec + cos 0 = tan 0. 2 J3 18. Tan o t coto 4. . CHAPTER V. LOGARITHMS. 20. DEF.-The logarithm of a number to a given base is the index of the power to which the base must be raised to obtain the number, Thus, we may obtain the numbers 1, 10, 100, 1,000, 10,000, &c., by raising the base 10 to the powers 0, 1, 2, 3, 4, &c., respectively; and hence, by the above definition, we have Log 1 = 0, log 10 = 1, log 100 = 2, log 1,000 2, log 1,000 = 3, &c., 10 10 10 10 the suffix 10 being added to the word log to indicate that the base is 10. It is usual, however, in common logarithms to omit this suffix; and hence, when there is no base expressed, the student will understand 10. Again, the numbers 1, 2, 4, 8, 16, &c., may be obtained by finding the values of 20, 21, 22, 23, 24, &c., respectively, and hence we have by definitionLog 1 = 0, log 2 = 1, log 4 = 2, log, a 3, &c. So also we find log 16 = 2, log 125 = , 3, log 81 4, &c. Ex. Find log 256, log 216, and the logarithm of 9 to base 13. Log 256 = log, 41 = 4, by definition. Log 216 = log 6* = log (89? = log 36* = }, by definition . Log 9 = log 38 = log (V3)* = 4, by definition. V3 13 4 5 3 36 Characteristics of Ordinary Logarithms. 21. DEF.-The characteristic of a logarithm is the integral part of the logarithm, and the fractional part (generally expressed as a decimal) is called the mantissa. 10n +13 1. Numbers containing integer digits. Every number containing n digits in its integral part must lie between 10n-1 and 10”, Thus, 6 lies between 10° and 101, 29 lies between 102 and 10%, 839 lies between 10% and 10%, &c. Hence the ordinary logarithms of all numbers having n integer digits lies between (n − 1) and n. The integral portion or characteristic of the logarithm of a number having n integer digits is therefore (n − 1). Hence we have the following rule : RULE 1.—The characteristic of the logarithm of a number having integer digits is one less than the number of integer digits. Thus, the characteristics of the logarithms of 32, 713.54, 8.7168, 56452, 73607.9 are respectively 1, 2, 0, 4, 4. 2. Numbers less than unity expressed as decimals. 1 1 decimal point lie between and or between 10-1 10n and 10 - In + 1). 1 Thus, 3 lies between 1 and .l, or between 1 and or 10° and 10' 10 - 1; 1 1 027 lies between 'I and .01, or between and or 10-1 10 and 10-?; 1 000354 lies between .001 and .0001, or between 10% ånd 100 or 10 – 3 and 10 – 4, and so on. Hence, by Def., Art. 20, the logarithm of any number having n zeros immediately after the decimal point lies between - n and - (n + 1). Hence, the logarithm is negative, and the integral part of this negative quantity is n. It is however usual to write all the mantissæ of logarithms as positive quantities, and the negative integral part of the logarithm will be the next higher negative integer, viz., - (n + 1). We have therefore the foHowing rule : RULE 2.—The characteristic of the logarithm of a number less than unity, and expressed as a decimal, is the negative 102 integer next greater than the number of zeros immediately after the decimal point. Thus, the characteristics of the logarithms of 3, •0076, ·02535, 7687, are respectively - 1, -3, - 2, – 1. 22. The logarithm of the PRODUCT of two numbers is the Sum of the logarithms of the numbers. Let m and n be the numbers, and let a be the base. Since m and n must be each some power of a, integral or fractional, positive or negative, assume man} Then, by definition of a logarithm, 2 = loga m, and y = loga n. Now we have mn = af.a = at+, and hence, by definition, log, mn = x + y; we therefore have (mn) log, m + loga n. Q.E.D. COR. This proposition may be extended to any number of factors. Thus, loga (mnpq) = loga m + logan + n + loga p + loga 9. 23. The logarithm of the QUOTIENT of two numbers is found by SUBTRACTING the logarithm of the denominator from the logarithm of the numerator. Assuming, as in the last Art., we have logam, y = logan. am Also, at - Y, and hence, by definition, ay a loga m n m loga a n log, m log, n. Q.E.D. 24. The logarithm of the POWER of a number is found by MULTIPLYING the logarithm of the number by the INDEX of the power. |