Computing Sine Coefficients


Similarly to how you approximated exe^x by finding the closest quadratic ax2+bx+cax^2 + bx + c, you must now find the best approximation of the form

f(x)≈s1sin⁡(x)+s2sin⁡(2x)+⋯+s5sin⁡(5x)f(x) \approx s_1 \sin(x) + s_2 \sin(2x) + \cdots + s_5 \sin(5x)

to each of the functions below.


Problem 1: f(x)=sin⁡(x)f(x)=\sin(x)

How well can you approximate sin⁡(x)\sin(x)? Use the sliders below, or enter the coefficients in their boxes. Check the boxes to see the individual component functions.

Approximate f(x) as closely as possible.
Approximate f(x) as closely as possible.

Was that easy for you?


Problem 2: f(x)=2sin⁡(2x)−8sin⁡(4x)f(x) = 2 \sin(2x) - 8 \sin(4x)

How well can you approximate 2sin⁡(2x)−8sin⁡(4x)2 \sin(2x) - 8 \sin(4x)? Use the sliders below, or enter the coefficients in their boxes. Check the boxes to see the individual component functions.

Approximate f(x) as closely as possible.
Approximate f(x) as closely as possible.

Still easy?


Problem 3: f(x)=3sin⁡(3x)+15sin⁡(5x)f(x)=3\sin(3x) + 15\sin(5x)

How well can you approximate 3sin⁡(3x)+15sin⁡(5x)3\sin(3x) + 15\sin(5x)? Use the sliders below, or enter the coefficients in their boxes. Check the boxes to see the individual component functions.

Approximate f(x) as closely as possible.
Approximate f(x) as closely as possible.

Not harder yet?


Problem 4: f(x)=sin⁡(15x)f(x)=\sin(15x)

How well can you approximate sin⁡(15x)\sin(15x)? Use the sliders below, or enter the coefficients in their boxes. Check the boxes to see the individual component functions.

Approximate f(x) as closely as possible.
Approximate f(x) as closely as possible.

Did this one get you?


Problem 5: The Sawtooth Function

How well can you approximate sin⁡(x)\sin(x)? Use the sliders below, or enter the coefficients in their boxes. Check the boxes to see the individual component functions.

Approximate f(x) as closely as possible.
Approximate f(x) as closely as possible.

This one had to take you some time.