Equation using sin and cos

Question

x1 y1

0 55

25 32.5

50 10

75 32.5

100 55

125 32.5

150 10

175 32.5

200 55

What are two equations that can be created from this relationship one equation using sin and one equation using cos?

Answer 1

To make things clearer, I'm going to use x instead of x1, and y instead of y1.

y as a function of x is definitely periodic. y has a maximum at y=0, so it looks like a cosine wave with period 100.

The cosine varies from a maximum of 55 to a minimum of 10, a range of 45. Half the range is 22.5, so that should be the amplitude of the cosine. The midpoint between the maximum of 55 and the minimum of 10 is 32.5, so that looks like the vertical offset of the wave.

y = Amplitude × cos(2πx/period) + Offset

Is y = 22.5 × cos(2πx/100) + 32.5?

x -> y = 22.5×cos(2πx/100)+32.5

x = 0 -> y = 22.5×cos(0)+32.5 = 22.5×1+32.5 = 55

x = 25 -> y = 22.5×cos(2π×25/100)+32.5 = 22.5×cos(π/2)+32.5 = 22.5×0+32.5 = 32.5

x = 50 -> y = 22.5×cos(2π×50/100)+32.5 = 22.5×cos(π)+32.5 = 22.5×(-1)+32.5 = 10

x = 75 -> y = 22.5×cos(2π×75/100)+32.5 = 22.5×cos(3π/2)+32.5 = 22.5×0+32.5 = 32.5

x = 100 -> y = 22.5×cos(2π×100/100)+32.5 = 22.5×cos(2π)+32.5 = 22.5×1+32.5 = 55

x = 125 -> y = 22.5×cos(2π×125/100)+32.5 = 22.5×cos(5π/2)+32.5 = 22.5×0+32.5 = 32.5

x = 150 -> y = 22.5×cos(2π×150/100)+32.5 = 22.5×cos(3π)+32.5 = 22.5×(-1)+32.5 = 10

Yes, y = 22.5 × cos(2πx/100) + 32.5