iterated double angle formula — speed of convergence
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The double angle formulas (1) and (2), combined with the best linear approximations at $0$ (3) and (4) can be used to estimate $sin$ or $cos$ by expanding an expression $n$ times and shrinking the angle to $frac{theta}{2^n}$ and then applying the linear approximation.
$$ sin(theta) = 2sin(theta cdot 1/2)cos(theta cdot 1/2) tag{1} $$
$$ cos(theta) = cos^2(theta cdot 1/2) - sin^2(theta cdot 1/2) tag{2} $$
$$ sin(theta) approx theta tag{3} $$
$$ cos(theta) approx 1 tag{4} $$
It seems to converge slowly, maybe adding a bit of accuracy or less per iteration.
How quickly does it converge to the true value of $sin(theta)$ ?
Here's a sample illustrating the relatively slow convergence.
$$
begin{array}[cc]
;n & sin_n(1) \
0 & 1.0 \
1 & 0.9375 \
2 & 0.89233017 \
3 & 0.86740446 \
4 & 0.8545371 \
5 & 0.84802556 \
6 & 0.84475327 \
7 & 0.8431133 \
end{array}
$$
trigonometry
asked Dec 3 at 4:02
Gregory Nisbet
410311
add a comment |
up vote
2
down vote
favorite
The double angle formulas (1) and (2), combined with the best linear approximations at $0$ (3) and (4) can be used to estimate $sin$ or $cos$ by expanding an expression $n$ times and shrinking the angle to $frac{theta}{2^n}$ and then applying the linear approximation.
$$ sin(theta) = 2sin(theta cdot 1/2)cos(theta cdot 1/2) tag{1} $$
$$ cos(theta) = cos^2(theta cdot 1/2) - sin^2(theta cdot 1/2) tag{2} $$
$$ sin(theta) approx theta tag{3} $$
$$ cos(theta) approx 1 tag{4} $$
It seems to converge slowly, maybe adding a bit of accuracy or less per iteration.
How quickly does it converge to the true value of $sin(theta)$ ?
Here's a sample illustrating the relatively slow convergence.
$$
begin{array}[cc]
;n & sin_n(1) \
0 & 1.0 \
1 & 0.9375 \
2 & 0.89233017 \
3 & 0.86740446 \
4 & 0.8545371 \
5 & 0.84802556 \
6 & 0.84475327 \
7 & 0.8431133 \
end{array}
$$
trigonometry
asked Dec 3 at 4:02
Gregory Nisbet
410311
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
The double angle formulas (1) and (2), combined with the best linear approximations at $0$ (3) and (4) can be used to estimate $sin$ or $cos$ by expanding an expression $n$ times and shrinking the angle to $frac{theta}{2^n}$ and then applying the linear approximation.
$$ sin(theta) = 2sin(theta cdot 1/2)cos(theta cdot 1/2) tag{1} $$
$$ cos(theta) = cos^2(theta cdot 1/2) - sin^2(theta cdot 1/2) tag{2} $$
$$ sin(theta) approx theta tag{3} $$
$$ cos(theta) approx 1 tag{4} $$
It seems to converge slowly, maybe adding a bit of accuracy or less per iteration.
How quickly does it converge to the true value of $sin(theta)$ ?
Here's a sample illustrating the relatively slow convergence.
$$
begin{array}[cc]
;n & sin_n(1) \
0 & 1.0 \
1 & 0.9375 \
2 & 0.89233017 \
3 & 0.86740446 \
4 & 0.8545371 \
5 & 0.84802556 \
6 & 0.84475327 \
7 & 0.8431133 \
end{array}
$$
trigonometry
asked Dec 3 at 4:02
Gregory Nisbet
410311
The double angle formulas (1) and (2), combined with the best linear approximations at $0$ (3) and (4) can be used to estimate $sin$ or $cos$ by expanding an expression $n$ times and shrinking the angle to $frac{theta}{2^n}$ and then applying the linear approximation.
$$ sin(theta) = 2sin(theta cdot 1/2)cos(theta cdot 1/2) tag{1} $$
$$ cos(theta) = cos^2(theta cdot 1/2) - sin^2(theta cdot 1/2) tag{2} $$
$$ sin(theta) approx theta tag{3} $$
$$ cos(theta) approx 1 tag{4} $$
It seems to converge slowly, maybe adding a bit of accuracy or less per iteration.
How quickly does it converge to the true value of $sin(theta)$ ?
Here's a sample illustrating the relatively slow convergence.
$$
begin{array}[cc]
;n & sin_n(1) \
0 & 1.0 \
1 & 0.9375 \
2 & 0.89233017 \
3 & 0.86740446 \
4 & 0.8545371 \
5 & 0.84802556 \
6 & 0.84475327 \
7 & 0.8431133 \
end{array}
$$
trigonometry
trigonometry
asked Dec 3 at 4:02
Gregory Nisbet
410311
asked Dec 3 at 4:02
Gregory Nisbet
410311
edited Dec 3 at 6:03
asked Dec 3 at 4:02
Gregory Nisbet
410311
asked Dec 3 at 4:02
Gregory Nisbet
410311
asked Dec 3 at 4:02
Gregory Nisbet
410311
410311
add a comment |
add a comment |
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