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xxxxx Problems and Quizzes related to Inverse Trigonometric Ratios Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IITJEE, Anyone else who needs this Tutorial as a reference!Solved Problems:
1. Problem : Find the principal value cot1(13) 2. Problem : Evaluate each of the following: (a) cos (Arcsin 3/5), (b) sin [Arccos (2/3)], (c) tan [Arcsin (3/4)] 3. Problem : Show that (i) sin1 ( 2x1x2) = 2 sin1x, 12≤x≤ 12 (ii) sin1 (2x1x2) = 2 cos1x, 12≤x≤1 4. Problem : Find the domain of sin^{1}2x+π/6. 5. Problem : Express tan1( cosx1sinx ), π2<x<π2 in the simplest form. 6. Problem : Find the value of sin1 (sin 3π/5) 7. Problem : Show that sin112/13 + cos14/5 + tan163/16 = π 8. Problem : Evaluate cos (Arctan 15/8  Arcsin 7/25). 9. Problem : Solve tan12x + tan13x = π/4 10. Problem : Prove that 2 tan113 + tan117 = π/4 11. Problem : Solve the equation tan1x+1x1 + tan1x1x = tan1(7) Solution : Here we havea + b = c wherea = tan1x+1x1 => tana = x+1x1 b = tan1x1x => tanb = x1x andc = tan1(7) => tanc = 7 Since tan(a+b) = tanc∴ tana+tanb1tana.tanb = tanc i. e. x+1x1+x1x1x+1x1.x1x = 7 i.e. 2x2x+11x = 7 so that x=2 This value makes the lefthand side of the given equation positive, so that there is no value of x strictly satisfying the given equation. The value x = 2 is a solution of the equation tan1x+1x1 + tan1x1x = π + tan1(7) 12. Problem : Solve Arccos 2x  Arccos x = π/3 . 13. Problem : Graph i) y = cos^{1}x + 1 ii) y = sin^{1}(x2) 14. Problem : Show that one can use a composition of trigonometry buttons such as, sin, cos, tan, sin−1, cos−1, and tan−1, to replace the broken reciprocal button on a calculator. 15. Problem : A camera is placed on a deck of a pool. A diver is 18 feet above the camera lens. The extended length of the diver is 8 feet. 16. Problem : If sin1 x + (sin1 y + sin1 z) = π/2, Find out x2 + y2 + z2 + 2xyz. 17. Problem : If sin1(x x22 + x34 ....) cos1(x2  x42 + x64  ....) = π/2 18. Problem : Solve the equation: tan1 2x + tan1 3x = nπ + π/4 19. Problem : Which angle is greater? A = 2tan1(22  1) and B = 3sin1(13) + sin1(35) Solution : We observe 22  1≈ 2(1.4) – 1 = 2.8 – 1 = 1.8So 22 – 1 > 3 => 2tan1(22 – 1) >2tan13 = 2π3 B = 3sin1(13) + sin1(35) = sin1[3. 13  4(13)3] + sin1(35) = sin1(2327) + sin1(35) < sin1(32) + sin1(32) = 2π3 [∵2327 < 32 and 35 < 3 2] So A > B.20. Problem : Suppose a calculator is broken and the only keys that still work are the sin, cos, tan, sin−1, cos−1, and tan−1 buttons. The display initially shows 0. Given any positive rational number q, show that we can get q to appear on the display panel of the calculator by pressing some finite sequence of buttons. Assume that the calculator does realnumber calculations with infinite precision, and that all functions are in terms of radians. Solution: Because cos−1 sin θ = π2−θ and tan(π2−θ)= 1tanθ for 0 < θ < π2 , we have for any x > 0, tan cos−1 sin tan−1 x = tan(π2− tan−1x)= 1x.............................. (*) Also, for x ≥ 0,cos tan−1√x = 1x+1 , so by (*),tan cos−1 sin tan−1 cos tan−1√x =x+1............................. (**) By induction on the denominator of r, we now prove that √r, for every nonnegative rational number r, can be obtained by using the operations√x → x+1 and x → 1x. If the denominator is 1, we can obtain √r, for every nonnegative integer r, by repeated application of √x → x+1. Now assume that we can get√r for all rational numbers r with denominator up to n. In particular, we can get any ofn+11, n+12, ... n+1n, so we can also get1n+1, 2n+1, ... nn+1, and√r, for any positive r of exact denominator n + 1, can be obtained by repeatedly applying √x → x+1. Thus for any positive rational number r, we can obtain √r. In particular, we can obtain q2 = q. Complete Tutorial (MCQ Quizzes after this):
MCQ Quiz #1Companion MCQ Quiz #1 for Inverse Trigonometric Ratios (Challenging Problems)  test how much you know about the topic. Your score will be emailed to you at the address you provide.MCQ Quiz #2Companion MCQ Quiz #2 for Inverse Trigonometric Ratios (Challenging Problems)  test how much you know about the topic. Your score will be emailed to you at the address you provide.
