Odd Angles Formula Trivia

08152019, 01:55 AM
(This post was last modified: 10082019 09:14 PM by Albert Chan.)
Post: #4




RE: Odd Angles Formula Trivia
Numerically, it is better not to use multiple angle formula for cos(n*x)
When x is small, cos(x) approach 1.0, and might "pushed away" many significant digits. Example, using HP12C, calculate cos(1), by applying T_{5} 3 times T_{5}(x) = 5x  20x^3 + 16x^5 = x(5  x²(20  x²(16))) x = 1/5³ = 0.008 // reduced angle, in radian cos(x) ≅ 1  x^2/2 + x^4/24 = 1  0.00003199982933 = 0.9999680002 cos(0.04) = 0.9992001048, error = 19 ULP cos(0.20) = 0.9800665309, error = 469 ULP cos(1.00) = 0.5403013120, error = 9939 ULP Had we use sin(5x) formula (same T_{5}, but start with sin(x)): sin(x) ≅ x  x^3/6 = 0.007999914667 sin(0.04) = 0.03998933419, error = 0 ULP sin(0.20) = 0.1986693308, error = 0 ULP sin(1.00) = 0.8414709849, error = 1 ULP cos(1) = √(1  sin(1)²) = 0.5403023058, error = 1 ULP Actual value for cos(1) = 0.54030 23058 68139 ... note: error (underestimated) = exact  approx 

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Messages In This Thread 
Odd Angles Formula Trivia  Albert Chan  12202018, 05:38 PM
RE: Odd Angles Formula Trivia  Albert Chan  12212018, 01:10 PM
RE: Odd Angles Formula Trivia  Albert Chan  08142019, 01:54 PM
RE: Odd Angles Formula Trivia  Albert Chan  08152019 01:55 AM
RE: Odd Angles Formula Trivia  Albert Chan  08152019, 03:42 PM
RE: Odd Angles Formula Trivia  Albert Chan  12152019, 01:03 AM
RE: Odd Angles Formula Trivia  Albert Chan  12152019, 01:49 AM
RE: Odd Angles Formula Trivia  Albert Chan  01222020, 12:45 AM
RE: Odd Angles Formula Trivia  Albert Chan  10132021, 03:40 AM

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