How Calculus helps us really understand Trigonometry

Compare Middle Ages bootstrapping of sine values using the unit angle and the combination formula with calculus' dynamic approach through Taylor series, analyzing how sine changes with angle and velocity.

Unpack how Taylor series expansion links initial velocity and acceleration to projected distance, using average extra velocity and velocity levels, with differentiation and integration techniques.

Explore how Taylor series and differentiation yield the first level velocity with respect to angle, in radians, revealing that instantaneous velocity equals cosine theta on a unit circle.

Derive velocity as cosine theta and acceleration as negative sine theta through successive velocity layers, noting a four-level cycle. Connect velocity, acceleration, and displacement with average velocity, hinting at integration.

Explore how breaking time into small intervals and summing constant-velocity distances leads to the integration concept, handling acceleration and limits to relate distance and time.

Explore how integration handles changing velocity across multiple lower levels, including acceleration, derive velocity at time t via the limit process, and compute distance by summing velocity over intervals.

Explore differentiation by deriving acceleration, velocity, and distance from a distance formula, and show how delta t and the limit yield instantaneous velocity, instantaneous acceleration, and average velocity.

Learn how to differentiate a general polynomial term using the binomial theorem, derive velocity as n t^{n-1}, and see the link to integration via the fundamental theorem of calculus.

Uncover how calculus explains trigonometry by proving that the rate of change of sine with respect to theta equals cosine theta, using limits and geometric reasoning.

Differentiate cosine by using the limit with delta theta to reveal the derivative is negative sine theta, then trace the four-level cycle of derivatives and connect to Taylor series expansion.

Derive sine and cosine via Taylor series by differentiating and integrating velocity levels to obtain initial terms, yielding the alternating sine series and the cosine series with even terms.

Implement Taylor series in Excel to compute sine and cosine from degrees to radians, using factorials and odd/even terms, and compare with functions for real life applications like Fourier analysis.

Explore Fourier analysis, a key tool in sound analysis, separating signals into independent sine-wave frequencies and revealing each frequency component.

Demonstrates creating sine waves for different frequencies, applies Fourier analysis, and explains how to wrap the Taylor series using a remainder modulo two pi to compute sine values accurately.

We use Taylor series to create sine waves in Excel for Fourier analysis, comparing same and different frequencies to show how they respond and why Fourier methods work.

Apply algebra to justify why Fourier analysis works by transforming sine products into cosines. See how frequency difference and addition govern summed signals, with amplitude scaling demonstrated in practice.

Learn to construct a composite sound signal from multiple sine waves and extract its amplitude and phase using sine cosine products, via Fourier analysis for sound analysis.

Learn how to compute binomial coefficients by counting selections and using factorials, and see how the binomial theorem applies to any two components a and b.

Explore the sine and cosine addition formulas, visualizing on the unit circle to break down sin(a+b) and cos(a+b) into sine and cosine components.