By G. A. Pratt, C. W. Schofield
ISBN-10: 0080117465
ISBN-13: 9780080117461
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Answers: 2. (i) x = ηπβ. 3. -sin3xsin2x; x = 0°, 36°, 72°, 108°, 144°, 180°, 22|°, 67i°, 112i°, 157i°. TRIGONOMETRICAL EQUATIONS Question 14 Solve the following equations, giving general solutions: (i) tanx + tan2x = 0 (ii) sinx + sin2x = sin §x (iii) sinx + cosx = l/\/2. , Summer 1960. Paper I, Qu. ·. tan2x = - t a n * = tan(Tr-x), 2x = ηπ + (π - x), 3x = (n + 1)π, kir x = -x-, on substituting fc for (« +1). e. (iii) X-gWT. )»£. 38 TRIGONOMETRICAL EQUATIONS 39 Notes (i) The solution given here is the easiest one.
Hints (a) Express a, b, c in terms of d and use x = a - b - c. tan Θ then emerges. (b) The question can also be solved using the sine rule but this is a longer method. Answer: 23° 57'. 2. Two right-angled triangles ABP9 ABQ lie in the same plane on opposite sides of their common hypotenuse AB. If AB = d, the angle PAB = a and the angle BAQ = jS, prove that PQ =dsin(a+ß). One of the triangles is revolved about AB until the planes of the triangles are perpendicular. Prove that now PQ =rffsin2(a + ß) - \ sin 2a sin 2j8]i \\ (London, January 1962.
E. > 9. 1 -(a +b + c) +(ab + bc + cd) -abc abc / l 1 1\ = I - + T + - ) - 1 using a + b + c = 1 > 9 - 1 using above result >8. Notes (a) In proving inequalities it is often helpful to use the fact that the square of a real expression cannot be negative. (b) Each result in (ii) is used in obtaining the next result. Practice Questions 1. The arithmetic mean (A) and geometric mean (G) of two positive numbers a, b are defined to be %(a + b)9 +\/ab respectively. Prove that G < A with equality if and only if a = b.