Three Trig ProblemsDate: 6/11/96 at 21:9:10 From: Anonymous Subject: Three trig problems I would like you to answer three trig problems that have been baffling me for a while. (1). Find tan x if ((sin x)^2)/3 + ((cos x)^2)/7 = (-sin(2x)+1)/10 (2). A = 20 deg. and B = 25 deg. Find the value of (1+tanA)(1+tanB). (3). Evaluate cos 36 - cos 72. - Thanks for your help. Michael Date: 6/12/96 at 7:55:2 From: Doctor Pete Subject: Re: Three trig problems Again, here are a few hints: 1. Note that the half-angle formulas give sin^2(x) = (1-cos(2x))/2, cos^2(x) = (1+cos(x))/2; substitute and simplify to obtain an expression in cos(2x) and sin(2x). Noting that cos(2x) = sqrt(1-sin^2(2x)), let y = sin(2x), and appropriate solving, using the fact that 20^2+21^2 = 29^2, will give sin(2x) = -21/29. 2. Since tan(A+B) = tan(45) = 1, consider tan(A) = tan((A+B)-B) and apply the tangent addition formula, resulting in (1+tan(A))/(1+tan(B)) = 2. 3. An indirect way of evaluating this is to think geometrically. If you are familiar with complex numbers, consider the complex 5th roots of unity in the plane; these form the vertices of a regular pentagon. Let the solutions to z^5-1 = 0 be z(0), z(1), ..., z(4), where z(0) = 1 and the remaining roots are counted counterclockwise. So we want to find the real part of z(1), which is cos(2*Pi/5) = cos(72). To do this, factor z^5-1 = (z-1)(z^4+z^3+z^2+z+1). The latter factor can be broken into two quadratics by considering the product (z^2+a*z+1)(z^2+b*z+1) (why can we restrict the coefficients of the square and constant terms to be 1?), and finding a and b such that the expansion of the above gives z^4+z^3+z^2+z+1. Since z(1) and z(4) are complex conjugates, their sum is twice the real part of z(1); but this is precisely the negative of the coefficient of the x term of one of these quadratic factors (which one, and why?). This will give you cos(72), and cos(36) is easily found using the half-angle formula; the answer is 1/2. There should be an easier way to do (3); the above hints are just solutions that come immediately to mind. If you have difficulty filling in the details (esp. on 3), feel free to ask. -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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