4. Which of the trigonometric functions are always less than unity? which always greater? which sometimes greater and sometimes less? 5. Write down the formulas for the sine and cosine of the sum, and the sine and cosine of the difference of two angles. 6. Prove the formula sin2 a + cos2 a = 1. 7. From the formulas of the two preceding questions deduce formulas for the sine, cosine, and tangent of twice an angle, and of half an angle. 8. To solve a triangle in which two sides and an angle opposite one of them are given. Example: one side = 47.6, another side = 32.9, and the angle opposite the latter side — 53° 24'. VI. 1. Prove that the logarithm of the product of several factors is equal to the sum of the logarithms of the factors 2. Work the following examples: (a.) 0.01706 X 8.7634 X 0-001 =? (&.)T.Trfm =? (ft) *£9 =? #029 =? (i) ^9 |pWP). -' Use arifchmetical com" plements in working the last. 3. Find the sines, cosines, and tangents, both natural and logarithmic, of the following angles: (A.) 24° 47' 22". 4. Prove the formula a2 — 62 + c2 — 2 be cos A. 5. Prove the formulas 1 + cos A = 2 (cos £ A)2. i — cos A = 2 (sin J A)2. 6. From the formulas of the last two questions deduce the formula sin J A = u& ^ ^ 7. The sides of a triangle are 37, 41, and 48; what are the angles? 8. To solve a triangle when two sides and the included angle are given. Example: Given the sides 47.6 and 58.4, the included angle 52° 24'. VII. 1. In a system of logarithms of which the base is 16, what is the number of which the logarithm is —1.25? In the system of which 10 is the base, why do the logarithms of two numbers composed of the same series of significant figures differ only in their characteristics? 2. Prove that the logarithm of the continued product of several numbers is equal to the sum of their logarithms. 3. Write (without proving) the formulas for the sine and cosine of the sum and of the difference of two angles; and prove the formula cos A + cos B — 2 cos \ (A + B) cos k (A-B). 4. Give the values of the sine, cosine, and tangent of 0°, 90°, 180°, 270°, 360°. Find the formula for cos (270° -if). 5. Given in a triangle b = 0.1072, c = 0.0625, C= 20° W. Solve completely. i/(0 07323)a 6. Find by logarithms the value of 0 35393 x 3700' 7. Given the cotangent of an angle equal to 2 ^ 2; find the other trigonometric functions, by computation. VIII. 1. What is the reason that, in the common system, the logarithms of two numbers consisting of the same series of significant figures differ only in their characteristics? 2. Write (without proving) the formulas for the sine and cosine of the sum and of the difference of two angles; and deduce those for the sine and cosine of the double of an angle and of the half of an angle. 3. Find, by means of formulas, the trigonometric functions of 30° and 60°. 4. Prove that, in any triangle, ——r = . . ,, 5. Solve the triangle in which a = 110.6, i = 56.7, C = 108° 24'. 6. Find, by logarithms, the value of the fraction y/(0.027919)i' (0.0010708)2 X 7.9' 1. Obtain a formula by which, when the sine of an angle is known, its cosine may be found. Also formulas for finding the tangent and cotangent of an angle, when the sine and cosine are given. 2. Obtain, by the formulas of the previous question, the trigonometric functions of 45°. 3. Prove that, in any triangle, the sines of any two angles are proportional to the opposite sides. 4. Solve the triangle in which two sides are 32.64 and 25.14, and the angle opposite the second side is 32° 48'. Are there two solutions to this problem? Why? 6. State the process and give the formulas by which, when two sides and the included angle of a triangle are known, the remaining parts can be obtained. X. 1. In the system of logarithms with six for its base, of what numbers will 3 and — 3 be the logarithms? What will be the index of the logarithm of 2000? 2. Find, by logarithms, the value of t y^JLj^iL 3. Show, by means of a diagram, what lines may be taken to represent the sine and the cosine of angles in each of the four quadrants of a circle, the radius of the circle being unity. Show also what are the algebraic signs of these same functions in the different quadrants. 4. Obtain formulas for the trigonometric functions of a negative angle. 5. In a right plane triangle, one side is 0.1426 and the opposite angle is 47° 29'. Solve the triangle. 6. Write the formulas for the sine and the cosine of the sum of any two angles; and obtain from them formulas for the sine and the cosine of the double angle. The sine of a certain angle is Find the trigonometric functions of double that angle. 7. Two sides of a plane oblique triangle are 16.49 and 21.37, and the included angle is 129° 37'. Find the other two angles. State the method of finding the remaining side. 8. One angle of a plane triangle is 30°, and an adjacent side is 12. What values of the side opposite the given angle will give two solutions to the triangle? What values will give only one? What values will give no solution? XL 1. Between what two integers docs the common logarithm of 327.8 lie? Give the reason for your answer. 2. Find, by logarithms, the value of X (f)8 X $824J. 3. In what quadrants may an angle be taken whose secant is 1.25? Obtain the corresponding values of the sine. 4. Find all the functions of (180° + y). 5. The hypothenuse of a right triangle is 0.3287, and one side is 0.1938. Solve the triangle. 6. By means of the formulas for the sine and the cosine of the sum of two angles, obtain the formula, tan (x + y) tan x -f- tan y 1 — tan x tan y 7. The three sides of a triangle are 1.328, 1.416, and 0.9388. Find the angles. XII. 1. In a certain system of logarithms the logarithm of 0.125 is—1.5. What is the base? 2. Find, by logarithms, the value of v'f ?(.0048659)*. 3. Of the following angles, which have a cosine equal to —0.5? a tangent equal to 1? a cosecant equal to —*J2? 45°; 120°; 225°; 240°; 315°; —240°; —315°; 600°. 4. If sin 5. In any triangle ABC, prove that a2 = 62 -f- c2 — 2bc cos A. 6. Solve the right triangle, given an angle 47° 48' 13", and the opposite side 0.043629. |