Mathematics Dictionary Dr. K. G. Shih

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Trigonometric Equations

Subjects

Read Symbol defintion Q01 | - Series of sin(x) Q02 | - Series of cos(x) Q03 | - Series of tan(x) Q04 | - Series of arcsin(x) Q05 | - Series of arccos(x) Q06 | - Series od arctan(x) Q07 | - Prove that sin(x) = x - (x^3)/3! + (x^5)/5! - .... Q08 | - Compare the series of sin(x) and sinh(x) Q09 | - Compare the series of cos(x) and cosh(x) Q10 | -

Answers

Q01. Series of sin(x) Series of sin(x) sin(x) = x - x^3/3! + x^5/5! - ...... Example : Prove that Lim[sin(x)/x] = 1 as x tends to zero sin(x)/x = (x - (x^3)/3! + (x^5)/5! - ....)/x. sin(x)/x = 1 - (x^2)/3! + (x^4)/5! - ..... Hence Lim[sin(x)/x] = 1 as x tends to zero Go to Begin Q02. Series of cos(x) Seirs of cos(x) cos(x) = 1 - x^2/2! + x^4/4! - ...... Go to Begin Q03. Series of tan(x) Series of tan(x) tan(x) = x + x^3/3 + 2*x^5/15 + 17*x^7/315 + ..... Example : Prove that Lim[tan(x)/x] = 1 as x tends to zero tan(x)/x = (x + (x^3)/3 + (2*x^5)/15 + ....)/x. tan(x)/x = 1 - (x^2)/3 + (2*x^4)/15 + ..... Hence Lim[tan(x)/x] = 1 as x tends to zero Go to Begin Q04. Seires of arcsin(x) Seires of arcsin(x) arcsin(x) = x + x^3/(2*3) + (1*3)*x^5/(2*4*5) + (1*3*5)*x^7/(2*4*6*7) + .... Go to Begin Q05. Series of arccos(x) Series of arccos(x) Go to Begin Q06. Series of arctan(x) Series of arctan(x) arctan(x) = x - x^3/3 + x^5/5 +- .... Application If x = 1, we have arctan(1) = pi/4 Hence pi/4 = 1 - 1/3 + 1/5 - 1/7 + .... The series is simple but it converges very slow Reference Computer mathematics by Dr shih Chapter 5 : The arctan(x) Chpater 6 : the story of pi Go to Begin Q07. Prove that sin(x) = x - (x^3)/3! + (x^5)/5! - .... The proof are given in Series Go to Begin Q08. Compare series of sin(x) and sinh(x) Series of sin(x) sin(x) = x - (x^3)/3! + (x^5)/5! - .... Series of sinh(x) sinh(x) = x + (x^3)/3! + (x^5)/5! + .... Go to Begin Q09. Compare series of cos(x) and cosh(x) Series of cos(x) cos(x) = 1 - (x^2)/2! + (x^4)/4! - .... Series of cosh(x) cosh(x) = 1 + (x^2)/2! + (x^4)/4! + .... Go to Begin Q10. Go to Begin

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