Easy Trigonometrical Equations. 40. As a further exercise in using the formulæ of Art. 29 and the numerical values of the functions of 45°, 60°, 30°, we shall now give some examples in trigonometrical equations. Example 1. Solve 4 cos 4-3 sec A. By expressing the secant in terms of the cosine, we have The student will be able to understand the meaning of the negative result in (2) after he has read Chap. VIII. This is a quadratic equation in which tane is the unknown quantity, and it may be solved by any of the rules for solving quadratic equations. From (2), tan 0= 5 ğ, a result which we cannot interpret at present. 41. When an equation involves more than two functions, it will usually be best to express each function in terms of the sine and cosine. From (2), cos 0 = −3, a result which must be rejected as impossible, because the numerical value of the cosine of an angle can never be greater than unity. [Art. 16.] EXAMPLES. IV. c. Find a solution of each of the following equations: 25. 2 sin tan 6+1=tan 6+2 sin 0. 26. 6 tan 0-5/3 sec 0+12 cot 0=0. 27. If tan +3 cot 0=4, prove that tan 6=1 or 3. 28. Find cote from the equation cosec2+cot2 8=3 cot 0. MISCELLANEOUS EXAMPLES. A. 1. Express as the decimal of a right angle (1) 25o 37' 6·4"; 2. Shew that (2) 63° 21' 36". sin A cos A tan A+cos A sin A cot A=1. 3. A ladder 29 ft. long just reaches a window at a height of 21 ft. from the ground: find the cosine and cosecant of the angle made by the ladder with the ground. 4. If cosec A= 17 find tan A and sec A. 5. Shew that cosec2 A-cot A cos A cosec A-1=0. 6. Reduce to sexagesimal measure (1) 17° 18' 75"; (2) 0003 of a right angle. 7. ABC is a triangle in which B is a right angle; if c=9, a=40, find b, cot A, sec A, sec C. 8. Which of the following statements are possible and which impossible? (1) 4 sin 0=1; (2) 2 sec 0=1; (3) 7 tan 0=40. 9. Prove that cos 0 vers (sec +1)=sin2 0. 10. Express sec a and cosec a in terms of cot a. 13. If m sexagesimal minutes are equivalent to n centesimal minutes, prove that m=54n. , prove that tan A+sec A=3, when A is an cot (90° - A) cot A cos (90° - A) tan (90° - A) = cos A. 16. PQR is a triangle in which P is a right angle; if PQ=21, PR=20, find tan Q and cosec Q. = 1-2 cos2 a. 17. Shew that (tan a - cot a) sin a cos a= 18. Find a value of which satisfies the equation (1) 3 sin 0-2 cos20; (2) 5 cot - cosec2 0-3. 21. Prove that 1+2 sec2 A tan2 A - sec1 A — tan1 A =0. 22. In the equation 8 sin2010 sin 0+3=0, shew that one value of 6 is impossible, and find the other value. 23. In a triangle ABC right-angled at C, prove that 24. If cot A=c, shew that c+c-1-sec A cosec A. CHAPTER V. SOLUTION OF RIGHT-ANGLED TRIANGLES. A 42. EVERY triangle has six parts, namely, three sides and three angles. In Trigonometry it is usual to denote the three angles by the capital letters A, B, C, and the lengths of the sides respectively opposite to these angles by the letters a, b, c. It must be understood that a, b, c are numerical quantities expressing the number of units of length contained in the three sides. B a 43. We know from Geometry that it is always possible to construct a triangle when any three parts are given, provided that one at least of the parts is a side. Similarly, if the values of suitable parts of a triangle be given, we can by Trigonometry find the remaining parts. The process by which this is effected is called the Solution of the triangle. The general solution of triangles will be discussed at a later stage; in this chapter we shall confine our attention to rightangled triangles. 44. From Euc. I. 47, we know that when a triangle is rightangled, if any two sides are given the third can be found. Thus in the figure of the next article, where ABC is a triangle rightangled at A, we have a2=b2+c2; whence if any two of the three quantities a, b, c are given, the third may be determined. Again, the two acute angles are complementary, so that if one is given the other is also known. Hence in the solution of right-angled triangles there are really only two cases to be considered: I. when any two sides are given; II. when one side and one acute angle are given. |