8. The vertical angle of a triangle is 120o, and the dif 9 ference of the sides is equal to of the base : find the other angles. L sin 12° 50′ = 9.3465794, L sin 12° 51'=9.3471336, log 2=3010300. log 3=4771213. 9. One of the angles of a plane triangle is 60°, and the side opposite is to the difference of the two sides including it as 9 is to 2: find the other angles. L cos 78° 54′ 10′′=9′2843730, L cos 78° 54' 20′′ = 9.2842656. log 3=4771213, 10. In a triangle a=19, b=1, A-B=90°: find C. Z tan 41° 59′ = 9·9541834, log 3='4771213, L tan 42o =9.9544374. 11. A person standing in the same plane with two vertical poles, and at a distance from the nearer equal to the distance a between them, sees their summits in the same direction. After walking in a straight horizontal line b feet towards the nearer pole he observes that the altitude of one summit is double that of the other. Determine the heights of the two summits. 12. From the deck of a ship which is sailing due North a lighthouse is observed due East, and the altitude of its summit is found to be 12° 26'. After the ship has sailed ten miles the lighthouse is again observed, and its altitude is found to be 7° 17'. Find how far the lighthouse was distant from the ship at the first observation. I sin 5° 9'8'9530996, L sin 7° 17'-91030373, L sin 19° 43'9'5281053, L sin 77° 34' 9.9896932, = log 71142 8521261, log 7·1143 8521322. 222. The equation sin x=a asserts that x is an angle of which the sine is a ; it is found convenient to be able to express this relation also in another way, in which a stands alone. The notation used is this, x= = sin-1a. Similarly x= =cos-1a expresses that x is an angle of which the cosine is a; and x=tan-1a expresses that x is an angle of which the tangent is a; and so on. 223. Any relation which has been established among Trigonometrical Ratios may be expressed by means of the inverse notation. Thus, for example, we know that this may be written cos 20=2 cos20-1; 20=cos-1 (2 cos2 0—1): suppose that cos 0=a, so that = cos-1a, suppose that sin=a, so that 0=sin-1a, and cos 0= √1—a2, 2 sin-1a=sin1 (2a √1 — a2). thus 224. Also any relation which is expressed in the inverse notation may be converted into a relation expressed in the ordinary notation. Thus, for example, suppose we have given that 2 tan-1 a= = tan-1 2a 1-a2; take the tangents of both sides; thus tan (2 tan-1a): 2a suppose that tan ̄1a=0, so that a=tan 0, 225. As an example of the inverse notation suppose we require the value of sin (sin-1a+cos-1b). Let sin 1a=A, and cos-1b=B; then the proposed expression becomes now sin (A+B) or sin A cos B+ cos A sin B, sin A=a, cos A = √(1—a2), cos B=b, sin B = √(1-b2); therefore sin (sin-1a+cos−1b)=ab+ √(1—a2) √√/(1 − b3). Examples, like the result just given, are often proposed -1 1 for exercise; but it should be remembered that sin and COS-1 П 2 1 2 both have an infinite number of values, and thus is merely one out of an infinite number of possible values of the left-hand member. See Chapter XVIII. 1. If sin 0=a, express in the inverse notation sin 30=3 sin 0-4 sin3 0. 2. If tan 0=a, express in the inverse notation 7. In any right-angled triangle, in which C is the right angle, 1 10. If tan-1x+tan¬13x=tan-11 find a. 2 5-1 MISCELLANEOUS EXAMPLES. 1. THE difference of two angles of an isosceles triangle is 20 grades: determine all the angles in degrees. 2. Find the simplest value of x from sin 4x=cos 5x. 3. Find sin x from the equation 4 sin x+3 cos x=5. 4. If sin x + cos x=sin Д+cos A, shew that sin x must be equal to sin A or to cos A. 5. Find the greatest value of sin x cos x. 6. Find the least value of tan x + cot x. 7. In a triangle sin2 C=sin2A + sin2B: shew that the triangle is right-angled. 8. One angle of a triangle exceeds the difference of the other two by 60 degrees, and exceeds the smaller of the other two by 50 grades: find the angles. 9. One of the angles of a plane triangle is 120o, and the sides including it are in the ratio of 4 to 1: shew that the cotangents of the other angles are 3√3 and √3 10. Shew that 11. Shew that log 75'6 1.8785218, find log 756 and log (0756). |