How to calculate $\sin(37°)$ with a Taylor approximation?

Question

How to calculate

$$\sin(37°)$$

with a Taylor approximation accurate to 3 decimal digits?

I know it is not a difficult question, but I have no answers of my book and so far I have only determined the Taylor approach of the umpteenth order and not approached or calculated anything in any sense.

Answer 1

The easiest approach may be to use the Taylor series $\sin x=x-{1\over6}x^3+{1\over120}x^5-\cdots$, since $37^\circ=37\pi/180\approx0.646\lt1$:

$$\sin(37^\circ)=\sin\left(37\pi\over180\right)\approx\left(37\pi\over180\right)-{1\over6}\left(37\pi\over180\right)^3+{1\over120}\left(37\pi\over180\right)^5$$

(noting that the next term in the alternating sum is considerably less than $1/5040\approx0.0002$). I wouldn't want to complete the decimal calculation by hand, but it's relatively straightforward with a calculator, even if you have to use an approximation like $\pi\approx3.1416$ on a pocket calculator that lacks a button for $\pi$ and only does arithmetic.

Answer 2

Hint:

• $ \sin (a+x) = \sin(a) + x \cos(a) + O(x^2) $

• $ a = 36^\circ = 2 \pi /10 $

• $ x = 1^\circ = 2 \pi /360 \approx 0.0175 $

• $ \sin(a) = \frac{\sqrt{10 - 2\sqrt{5}}}{4}$, $\cos(a)=\frac{1+\sqrt{5}}{4}$ (see Wikipedia)

Answer 3

Expand around $\pi/5$ $$\sin x = \sin(\pi/5)+\cos(\pi/5) \left(x-\frac{\pi }{5}\right)-\frac{1}{2} \sin(\pi/5) \left(x-\frac{\pi }{5}\right)^2+\\+\frac{1}{6} \cos(\pi/5) \left(x-\frac{\pi }{5}\right)^3+\frac{1}{24} \sin(\pi/5)\left(x-\frac{\pi }{5}\right)^4+O\left(x^5\right)$$ You get $0.601$ plugging $x=37/180 \pi$

Answer 4

For $\alpha=\frac{37\cdot\pi}{180}=0.65477$ we take 3 terms in the Maclaurin series: $$ \sum_{k=0}^2 \frac{(-1)^k 0.64577^{1 + 2 k}}{(1 + 2 k)!}=0.64577-\frac{0.65477^3}{6}+\frac{0.65477^5}{120}=0.601823 $$ This is as close as $10^{-5}$ to the exact value.

Answer 5

Using Taylor's poloynomial with degree $p$ around $a=\frac{\pi}{6}$, the error is bounded by $$ |R_p| =\left| \dfrac{f^{(p+1)}(\xi)}{(p+1)!}\left(\frac{37 \pi}{180}-\frac{\pi}{6}\right)^{p+1}\right|\leq \frac{1}{(p+1)!} \left(\frac{7 \pi}{180}\right)^{p+1} $$ so, just select a convenient $p$ and use the corresponding Taylor polynomial to get the approximation.

For $p=3$ you get na error bound of $ 0.928304 \times 10^{-5}$ and for $p=4$ na error bound of $0.226828\times 10^{-6}$

Answer 6

I agree with the other answers that are already provided. However, I think that there is a possible source of confusion that should be resolved.

In Trigonometry/Analytical Geometry, the domain of the sine and cosine functions are angles, which have a unit of measurement, the degree. So $37^{\circ}$ is a dimensioned number in the same way that 1 foot is a dimensioned number.

In Calculus/Real Analysis, this (necessarily) changes. The domain of the sine and cosine functions are dimensionless real numbers (i.e. $\pi/4$ rather than $45^\circ$). The reason for this alteration in the sine and cosine functions is to facilitate (for examples)

• Using the Taylor series of the sine and cosine functions to attack problems

• Using the ArcTan function to attack a problem like $\int_0^1 \frac{dx}{1 + x^2}.$

Some of the confusion centers around ambiguity in the connotation of the term radian. That is, does $(\pi/4)$ radians represent a dimensioned number, with 1 radian representing a unit of measurement of an angle similar to what 1 degree represents? This connotation is often useful in allowing the Trigonometry/Analytical Geometry student to transition into Calculus/Real Analysis and immediately begin solving Calculus problems without requiring that the student first acquire a deep understanding of the theory.

The alternative connotation of 1 radian, is that it is used to represent the dimensionless proportion of a specific arc length of a portion of the unit circle. This proportion (i.e. ratio) is taken against the arc length of 1 complete revolution of the unit circle (i.e. the circumference of the unit circle). The circumference of the unit circle is $2\pi.$

Therefore, the term $(\pi/4)$ radians for example, can be interpreted in two totally distinct ways:

1. equivalent to $45^\circ$.

2. representing the arc length of 1/8-th of a complete revolution around the unit circle. This interpretation preserves the idea that $(\pi/4)$ radians (for example) is a dimensionless number.

Often, when a transitioning student uses the term radians, they are intending the first connotation above. When someone experienced in Calculus/Real Analysis uses the term radians, they are intending the second connotation above.

For an illustration of the sine and cosine functions having as their domain, dimensionless numbers, see What is the physical meaning of sine, cosine and tangent of an obtuse angle?.