8. Find L cot 36° 26′ 16′′, given L cot 36° 27'10-1315840, L cot 36° 26'=10·1318483. 9. Find L cos 55° 30′ 24′′, given L cos 55° 31'=9.7529442, L cos 55° 30′ = 9.7531280. 10. Find the angle whose tabular logarithmic sine is 9-8440018, using the data of example 7. 11. Find the angle whose tabular logarithmic cosine is 9.7530075, using the data of example 9. 12. Given L tan 24° 50′ = 9.6653662, diff. for 1'=3313, find Z tan 24° 50′ 52.5′′. 13. Given L cosec 40° 5′-10-1911808, diff. for 1'=1502, find L cosec 40° 4' 17.5". 182. Considerable practice in the use of logarithmic Tables will be required before the quickness and accuracy necessary in all practical calculations can be attained. Experience shews that mistakes frequently arise from incorrect quotation from the Tables, and from clumsy arrangement. The student is reminded that care in taking out the logarithms from the Tables is of the first importance, and that in the course of the work he should learn to leave out all needless steps, making his solutions as concise as possible consistent with accuracy. Example 1. Divide 6-6425693 by 3873007. Example 2. The hypotenuse of a right-angled triangle is 3·141024 and one side is 2.593167; find the other side. Let c be the hypotenuse, a the given side, and a the side required; then [In this exercise the logarithms are to be taken from the Tables.] 1. Multiply 300-2618 by 0078915194. 2. Find the product of 235 6783 and 357·8438. 3. Find the continued product of 153-2419, 2.8632503, and 07583646. 4. Divide 10304051 by 27-093524. 5. Divide 357-8364 by 00318973. 6. Find a from the equation 0178345x=21.85632. 7. Find the value of 3.78956 x 0536872÷0072916. H. K. E. T. 11 8. Find the cube of 83410039. 9. Find the fifth root of 15063-018. 10. Evaluate 384-731 and 15-7324. 11. Find the product of the square root of 1034·3963 and the cube root of 353246. 12. Subtract the square of 7503269 from the square of 1.035627. 13. Find the value of (34.7326) × √2·53894 Example 3. Find a third proportional to the cube of 3172564 and the cube root of 23-32873. Let x be the required third proportional; then (·3172564)3 : (23·32873)* = (23·32873)† ; x; x=(23.32873)+(-3172564); : whence 14. Find a mean proportional between 0037258169 and 56301078. 15. Find a third proportional to the square of 43607528 and the square root of '03751786. 16. Find a fourth proportional to 56712-43, 29-302564, 33025107. 17. Find the geometric mean between (035689) and (2.879432). 18. Find a fourth proportional to sin 20° 13′ 20′′ x cot 47° 53′ 15′′ x sec 42° 15′ 30′′. 22. Find the value of ab sin C, when a=324·1368, b=417·2431, C=113° 14′ 16′′. 23. If a b=sin A : sin B, find a, given b=378.25, A=35° 15′ 33′′, B=119° 14′ 18′′. 24. Find the smallest values of which satisfy the equations 5 (1) tan3 (2) 3 sin2 0+2 sin 0=1. 12 25. Find x from the equation xx sec 28° 17′ 25′′=sin 23° 18′ 5′′ x cot 38° 15′ 13′′. 26. Find from the equation sin3 0=cosa cot ß, where a 32° 47′ and 8=41° 19' CHAPTER XVI. SOLUTION OF TRIANGLES WITH LOGARITHMS. 183. The examples on the solution of triangles in Chap. XIII. furnish a useful exercise on the formulæ connecting the sides and angle of a triangle; but in practical work much of the labour of arithmetical calculation is avoided by the use of logarithms. We shall now shew how the formulæ of Chap. XIII. may be used or adapted for use in connection with logarithmic Tables. 184. To find the functions of the half-angles in terms of the sides. s 2bc .. 2 sin? 4-4 (8-c) (8-b) _ 2 (s—b) (s—c); 2bc = (s—b) (s - c) bc |