15. Prove the identities: (1) sine sec30+ √cose cosec30= √sec30 cosec30. (secA + cosecA) (secA cosecA − 1). (4) sin1A + cos1A + 2 sin2A cos2A = (sin2A + cos2A)2 = 1. (5) sec2A tanA + cosec2A cotA = secA cosecA (sec2A cosec2A-2). (6) sec1A tanA + cosec1A cotA = sec3A cosec3A (sec2A cosec2A-3). (7) (tan A-cotA)2 = sec2A cosec2A - 4. 16. Trace the changes in the value of sinA + cosA, as A increases from 0° up to 90°. 17. Trace the changes in the value of sinA cosA, as A increases from 0° up to 90°. 4 6. If s denote the sine, c the cosine, and t the tangent of any angle, show that 7. Prove the identities: (1) cosA cot A cosec3A -sinA tan2A sec3A secA cosecA (cosec2A — sec2A) (cosec2A sec2A − 2). (2) secoA (1-sinĜA) – 1 = 3 tan2A (1 + tan2A.). 8. Solve the simultaneous equations tan2A + cot2B = 4 sin2A + cos2B=11. CHAPTER III. APPLICATIONS OF TRIGONOMETRICAL RATIOS. 26. In this chapter we give some simple applications of trigonometrical ratios. We begin by showing how the trigonometrical ratios of angles enable us to combine in equations the angles and sides of right-angled triangles. 27. Relations between the Angles and Sides of Rightangled Triangles. Let ABC be a triangle in which the angle ACB is a right angle. It is usual in trigonometry to denote the A4 angles BAC, CBA, and ACB of any B 12 triangle ABC by the letters A, B, and C; while the letters a, b, and c are used to denote the numbers which represent the lengths of the sides BC, CA, and AB opposite these angles. In the triangle we are considering C=90°, and therefore by the definitions Hence we can express the side a in terms of the other sides and the trigonometrical ratios of A and B in the following four different ways: a = b tanA = b cotB = c sinA Similarly with the sides b and c: =c cosB. b=a tanB=a cotA= c sinB=c cosA. c = a secB = a cosecA = b secA = b cosecB. Ex. 1. ABC is a triangle in which the angle C is a right angle, the angle A = 60°, and the length of AC is 10 feet. What are the lengths of BC and AB? Ex. 2. ABC is a triangle right-angled at C. On the sides BC, CA squares BCDE, ACFG are described. From the points D, E, F, G, perpendiculars DH, EK, FL, GM are drawn to AB produced both ways. Show that (1) AM = BK, (2) DH-FL=EK - GM. Using the notation of this article for the angles and sides of DHAD sinDAH =(CD + CA) sin DAH Again and therefore DH-FL=(a+b) (sinA - sinB) Also and therefore Therefore FL=BF sinFBL=(a+b) sinB; = (c sinA + c sinB) (sinA – sinB) EKBE sinEBK = a sinA = c sinA × sinA = c sin2A, GM AG sinGAM=b sinB = c sinB x sinB = c sin2B; EK-GM c (sin2A - sin2B). DH-FL=EK - GM. EXAMPLES VIII. 1. ABC is a triangle right-angled at C and the angle A = 45°. If the length of AC is 40 feet, what are the lengths of AB and BC? 2. In a triangle ABC the angle C is a right angle, the angle B=30°, and the length of the side BC is 30 feet; find to the nearest inch the lengths of the sides AC and AB. 3. ABC is a triangle right-angled at C, and CD is drawn perpendicular to AB the hypotenuse. Show that CD a cosA = b cosB = c cosA cosB. 4. ABC is a triangle right-angled at C, and a system of lines is drawn as follows:-CD is drawn perpendicular to AB, DE perpendicular to CA, EF perpendicular to AB, FG perpendicular to CA, &c. Show that CD = a cosA, DE = a cos2A, EF = a cos3A, FG = a cos1A, &c. 5. TA and TB are two tangents to a circle whose centre is O and whose radius is r. Show that the radii of the two circles which can be drawn to touch this circle and the two tangents TA and TB are r 1- sina and r 1 + sina where a is the angle OTA. If r1 and r2 denote these two radii, r1 r1⁄2= r2. 6. AB, the side of a square, is of length 2a. From C, the middle point of AB, COD is drawn perpendicular to AB. From O, the centre of the square, OP and OQ are measured each equal to c along OC and OD respectively, and PA and QB are joined. If the angles PAC and QBC be denoted respectively by A and B, show that and that α =1-tanAtanB-1, (tanB–tanA)(tanB–tan’A)=8(1–tanA tanB). 28. Heights and Distances. One of the most important applications of trigonometry is the determination of the heights and distances of objects which |