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Trigonometric Identities Grade 12 Pdf Download

 
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MessagePosté le: Jeu 1 Sep - 16:05 (2016)    Sujet du message: Trigonometric Identities Grade 12 Pdf Download Répondre en citant




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tan 2(x) - sin2(x) = sin2(x) / cos2(x) - sin2(x) = [ sin2(x) - cos2(x) sin2(x) ] / cos2(x) = sin2(x) [ 1 - cos2(x) ] / cos2(x) = sin2(x) sin2(x) / cos2(x) = sin2(x) tan2(x) which is equal to the right hand side of the given identity. Hence cos(2x) cos(x) - sin(2x) sin(x) = 0 is equivalent to cos(3x) = 0 Solve for 3x to obtain: 3x = pi/2, 3x = 3Pi/2, 3x = 5pi/2, 3x = 7pi/2, 3x = 9pi/2 and 11pi/2 solutions: x = pi/6, pi/2, 5pi/6, 7pi/6, 3pi/2 and 11pi/6 Use the identities sin(2x) = 2 sin(x) cos(x) and cos(2x) = 1 - 2 sin2(x) to rewrite the given equation as follows given equation ( sin(2x) - cos(x) ) / ( cos(2x) + sin(x) - 1 ) = 0 ( 2 sin(x) cos(x) - cos(x) ) / ( 1 - 2 sin2(x) + sin(x) - 1) = 0 cos(x)( 2 sin(x) - 1 ) / [ - sin(x)( 2 sin(x) - 1 ) ] = 0 Divide numerator and denominator by 2 sin(x) - 1 to simplify; assuming that 2 sin(x) - 1 is not equal to zero. - cos(x) / sin(x) = 0 -cot(x) = 0 solutions: x = pi/2 and x = 3pi/2 We now need to verify that both solutions found make neither the denominator nor 2 sin(x) - 1 equal to zero. Use the identities cos(2x) = 2 cos2(x) - 1 and sin(2x) = 2 sin(x) cos(x) in the left hand side of the given identity. Trigonometry Problems and Questions with Solutions - Grade 12 . Solutions to the Above Problems Use the identity tan(x) = sin(x) / cos(x) in the left hand side of the given identity.

[ 1 + cos(x) + cos(2x) ] / [ sin(x) + sin(2x) ] = [ 1 + cos(x) + 2 cos2(x) - 1 ] / [ sin(x) + 2 sin(x) cos(x) ] = [ cos(x) + 2 cos2(x) ] / [ sin(x) + 2 sin(x) cos(x) ] = cos(x) [1 + 2 cos(x)] / [ sin(x)( 1 + 2 cos(x) ) ] = cot(x) Use the identity sin(2x) = 2 sin(x) cos(x) to write sin(4x) = 2 sin(2x) cos(2x) in the right hand side of the given identity. 2 sin(x / 2) cos(x / 2) + sin(x / 2) = 0 sin(x/2) [ 2 cos(x/2) + 1 ] = 0 factor which gives sin(x/2) = 0 or 2 cos(x/2) + 1 = 0 sin(x / 2) = 0 leads to x / 2 = 0 or x / 2 = Pi which leads ro x = 0 or x = 2pi 2 cos(x/2) + 1 = 0 leads to cos(x/2) = -1/2 which leads to x/2 = 2pi/3 and x/2 = 4pi/3 (the second solution leads to x greater than 2pi) solutions: x = 0, x = 4pi/3 and x = 2pi The given equation is already factored (2sin(x) - 1)(tan(x) - 1) = 0 which means 2sin(x) - 1 = 0 or tan(x) - 1 = 0 sin(x) = 1/2 or tan(x) = 1 equivalent equations to the above solutions: x = pi/6, 5pi/6, x = pi/4 and x = 5pi/4 Note that cos(2x + x) = cos(2x) cos(x) - sin(2x) sin(x) using the formula for cos(A + B). Use the identity sin(2x) = 2 sin(x) cos(x) to write sin(x) as sin(2 * x/2) = 2 sin(x / 2) cos(x / 2) in the right hand side of the given equation. sin(4x) / cos(2x) = 2 sin(2x) cos(2x) / cos(2x) = 2 sin(2x) = 2 [ 2 sin(x) cos(x)] = 4 sin(x) cos(x) which is equal to the right hand side of the given identity. Free Mathematics Tutorials Math and Precalculus Math Problems Algebra Questions and Problems Graphs of Functions, Equations, and Algebra Free Math Worksheets to Download Analytical Tutorials Solving Equation and Inequalities Online Math Calculators and Solvers graphing Free Graph Paper Math Software Applied Math The Applications of Mathematics in Physics and Engineering Antennas Exercises de Mathematiques Utilisant les Applets Calculus Calculus Tutorials and Problems Calculus Questions With Answers Free Calculus Worksheets to Download Free Calculus Worksheets to Download Geometry Geometry Tutorials and Problems Online Geometry Calculators and Solvers Free Geometry Worksheets to Download Trigonometry Trigonometry Tutorials and Problems for Self Tests Free Trigonometry Questions with Answers Free Trigonometry Worksheets to Download More Statistics Elementary Statistics and Probability Tutorials and Problems Mathematics pages in French Site About the author Download E-mail Primary Math Middle School Math High School Math Free Practice for SAT, ACT and Compass Math tests Trigonometry Problems and Questions with Solutions - Grade 12 Grade 12 trigonometry problems and questions with answers and solutions are presented. Round your answer to 3 significant digits. 403 Forbidden.. Prove the indentity tan2(x) - sin2(x) = tan2(x) sin2(x) Prove the indentity (1 + cos(x) + cos(2x)) / (sin(x) + sin(2x)) = cot(x) Prove the indentity 4 sin(x) cos(x) = sin(4x) / cos(2x) Solve the trigonometric equation given by sin(x) + sin(x/2) = 0 for 0 ≤ x ≤ 2 pi Solve the trigonometric equation given by (2sin(x) - 1)(tan(x) - 1) = 0 for 0 ≤ x ≤ 2 pi Solve the trigonometric equation given by cos(2x) cos(x) - sin(2x) sin(x) = 0 for 0 ≤ x ≤ 2 pi Solve the trigonometric equation given by ( sin(2x) - cos(x) ) / ( cos(2x) + sin(x) - 1 ) = 0 for 0 ≤ x ≤ 2 pi Prove that sin(105o) = ( sqrt(6) + sqrt(2) ) / 4 If sin(x) = 2/5 and x is an accute angle, find the exact values of a) cos(2x) b) cos(4x) c) sin(2x) d) sin(4x) Find the length of side AB in the figure below.

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