{Use of Tech} Newton's derivation of the sine and arcsine series Newton discovered the binomial series and then used it ingeniously to obtain many more results. Here is a case in point.
a. Referring to the figure, show that x = sin s or s = sin ⁻¹ x.
b. The area of a circular sector of radius r subtended by an angle θ is 1/2r²θ. Show that the area of the circular sector APE is s/2, which implies that
s = 2 ∫₀ˣ √(1 − t²) dt − x √(1 −x²)
c. Use the binomial series for f(x) = √(1 − x²) to obtain the first few terms of the Taylor series for s=sin ⁻¹ x.
d. Newton next inverted the series in part (c) to obtain the Taylor series for x=sin s. He did this by assuming sin s = ∑ aₖ sᵏ and solving x = sin(sin ⁻¹ x) for the coefficients aₖ. Find the first few terms of the Taylor series for sin s using this idea (a computer algebra system might be helpful as well).

Verified step by step guidance

1. Step 1: Understand the relationship between the variables in the figure. Since s is the angle subtended by the arc on the unit circle and x is the length of the vertical segment corresponding to sin s, we have the fundamental trigonometric identity: \(x = \sin s\). Consequently, this implies \(s = \sin^{-1} x\) because the arcsine function is the inverse of the sine function on the appropriate domain.
2. Step 2: Use the formula for the area of a circular sector. The area of a sector with radius \(r\) and angle \(\theta\) is given by \(\frac{1}{2} r^2 \theta\). For a unit circle (\(r=1\)), the area of sector APE is \(\frac{s}{2}\). Next, express this area in terms of an integral involving \(x\) and \(t\) by considering the area under the curve \(y = \sqrt{1 - t^2}\) from \$0\( to \)x\(. This leads to the equation: \)s = 2 \int_0^x \sqrt{1 - t^2} \, dt - x \sqrt{1 - x^2}$.
3. Step 3: Expand the function \(f(x) = \sqrt{1 - x^2}\) using the binomial series. Recall the binomial series expansion for \((1 + u)^p\) is \(\sum_{n=0}^\infty \binom{p}{n} u^n\), where \(\binom{p}{n} = \frac{p(p-1)\cdots(p-n+1)}{n!}\). Here, set \(u = -x^2\) and \(p = \frac{1}{2}\). Write out the first few terms explicitly to approximate \(\sqrt{1 - x^2}\).
4. Step 4: Substitute the binomial series expansion of \(\sqrt{1 - t^2}\) into the integral expression for \(s\). Integrate term-by-term from \$0\( to \)x\( to find the series expansion for \)\int_0^x \sqrt{1 - t^2} \, dt\(. Then, use this result along with the term \)x \sqrt{1 - x^2}\( to write the Taylor series for \)s = \sin^{-1} x$ up to the desired number of terms.
5. Step 5: To find the Taylor series for \(x = \sin s\), assume a power series form \(\sin s = \sum_{k=1}^\infty a_k s^k\). Substitute \(s = \sin^{-1} x\) from the series found in Step 4 into this assumed series and equate it to \(x\). By matching coefficients of powers of \(x\), solve for the coefficients \(a_k\). This inversion process recovers the Taylor series for \(\sin s\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, like arcsine, reverse the effect of trigonometric functions. For example, if x = sin s, then s = sin⁻¹ x. Understanding this relationship is crucial for expressing angles in terms of their sine values and for manipulating integrals and series involving these functions.

Area of a Circular Sector and Integral Representation

The area of a circular sector with radius r and angle θ is given by (1/2)r²θ. This geometric fact helps relate the angle s to an integral involving √(1 − t²), connecting geometry with calculus. Recognizing this allows the derivation of integral formulas for inverse trigonometric functions.

Binomial Series and Taylor Series Expansion

The binomial series generalizes the expansion of expressions like (1 + x)^n for any real exponent n. Applying it to f(x) = √(1 − x²) enables the derivation of a Taylor series for arcsine. Inverting this series yields the Taylor series for sine, illustrating how series expansions can be manipulated to find function representations.