103 trigonometry problems titu andreescu & zuming feng

Đây là cuốn sách tiếng anh trong bộ sưu tập "Mathematics Olympiads and Problem Solving Ebooks Collection",là loại sách giải các bài toán đố,các dạng toán học, logic,tư duy toán học.Rất thích hợp cho những người đam mê toán học và suy luận logic.

[...]... function We can also use the natural logarithm function to change products into sums, and then apply Jensen’s inequality This technique will certainly be helpful in solving problems 18 103 Trigonometry Problems such as Introductory Problems 19(b), 20(b), 23(a) and (d), 27(b), and 28(b) and (c) Because the main goal of this book is to introduce techniques in trigonometric computation rather than in functional... follows, where |P Q| denotes the length of the line segment P Q: |P Q| , |OP | |OQ| cos θ = , |OP | |P Q| , tan θ = |OQ| sin θ = |OP | , |P Q| |OP | sec θ = , |OQ| |OQ| cot θ = |P Q| csc θ = 2 103 Trigonometry Problems First we need to show that these functions are well defined; that is, they only depends on the size of θ, but not the choice of P Let P1 be another point lying on ray OA, and let Q1 be... = 60◦ and B = 30◦ (Figure 1.3, right) We reflect A across line BC to point D By symmetry, D = 60◦ , so triangle ABD is equilateral Hence, |AD| = |AB| and |AC| = |AD| Because ABC is a right 2 4 103 Trigonometry Problems triangle, |AB|2 = |AC|2 + |BC|2 So we have |BC|2 = |AB|2 − √ √ 3|AB| It follows that sin 60◦ = cos 30◦ = 23 , 2 √ √ cot 60◦ = 33 , and tan 60◦ = cot 30◦ = 3 or |BC| = |AB|2 4 sin 30◦... formulas for the sine, cosine, and tangent functions for angles in a restricted interval In a similar way, we can develop an addition formula for the cotangent function We leave it as an exercise 6 103 Trigonometry Problems By setting α = β in the addition formulas, we obtain the double-angle formulas sin 2α = 2 sin α cos α, cos 2α = cos2 α − sin2 α, tan 2α = 2 tan α , 1 − tan2 α where for abbreviation,... 4 and |CD| = 10 Suppose that lines AC and BD intersect at right angles, and that lines BC and DA, when extended to point Q, form an angle of 45◦ Compute [ABCD], the area of trapezoid ABCD 8 103 Trigonometry Problems Q A B P C D Figure 1.8 Solution: Let segments AC and BD meet at P Because AB CD, triangles |AB| ABP and CDP are similar with a side ratio of |CD| = 2 Set |AP | = 2x and 5 |BP | = 2y... , cot β + cot γ or cot β + cot γ = cot β cot γ − 1 Solving the system of equations 2 cot γ = 1 + cot β, cot β + cot γ = cot β cot γ − 1 or 2 cot γ = cot β + 1, cot γ (cot β − 1) = cot β + 1 10 103 Trigonometry Problems for cot β gives (cot β +1)(cot β −1) = 2(cot β +1) It follows that cot 2 β −2 cot β − 3 = 0 Factoring the last equation as (cot β − 3)(cot β + 1) = 0 gives cot β = 3 Thus [ABC] = 50 tan... that sin(θ + 90◦ ) = cos θ, cos(θ + 90◦ ) = − sin θ, sin(θ − 90◦ ) = − cos θ, cos(θ − 90◦ ) = sin θ, cos(−θ ) = cos θ, sin(−θ ) = − sin θ, sin(180◦ − θ ) = sin θ, cos(180◦ − θ ) = − cos θ 12 103 Trigonometry Problems Furthermore, by either reflecting A across the line y = x or using the second and third formulas above, we can show that sin(90◦ − θ ) = cos θ and cos(90◦ − θ ) = sin θ This is the reason... cos α sin x + sin α cos x = √ which is solvable in x if and only if −1 ≤ 2 ≤ m2 + n2 √ m2 +n2 m2 + n 2 , ≤ 1, that is, if and only if Setting m = a, n = 1, and = c gives the desired result 14 103 Trigonometry Problems (b) By the relations a 2 + 1 = (sin2 x + cos2 x)(a 2 + 1) = (sin2 x + 2a sin x cos x + a 2 cos2 x) + (a 2 sin2 x − 2a sin x cos x + cos2 x) = (sin x + a cos x)2 + (a sin x − cos x)2 ,... reader might want to match the functions y = sin 3x, y = 2 cos x , 3 3 y = 3 sin 4x, y = 4 cos(x − 30◦ ), y = 2 sin x − 3, and y = 2 sin[3(x + 40◦ )] + 5 2 with the curves shown in Figure 1.15 16 103 Trigonometry Problems 15 10 5 -500 -400 -300 -200 -100 100 200 300 400 -5 -10 -15 Figure 1.15 We leave it to the reader to show that if a, b, c, and d are real constants, then the functions y = a cos(bx+c)+d,... triangle ABC angles CAB, ABC, BCA of triangle ABC circumradius and inradius of triangle ABC area of region F area of triangle ABC length of line segment BC the arc of a circle between points A and B 103 Trigonometry Problems 1 Trigonometric Fundamentals Definitions of Trigonometric Functions in Terms of Right Triangles Let S and T be two sets A function (or mapping or map) f from S to T (written as f : S → . Cataloging-in-Publication Data Andreescu, Titu, 195 6- 103 trigonometry problems : from the training of the USA IMO team / Titu Andreescu, Zuming Feng. . ISBN 0-8 17 6-4 33 4-6 (acid-free paper) 1. Trigonometry Problems, exercises, etc. I. Title: One hundred and three trigonometry problems. II. Feng, Zuming.