
Ajay F. answered 10/26/23
College-Level Math Instructor Specializing in Calculus & Probability
While we might not have a good guess for sin(55°) = sin(55° π / 180°) = sin(11π/36), we do know that sin(60°) = sin(π/3) = sqrt(3)/2. Since d/dx(sin(x)) = cos(x), the value of the derivative at x=π/3 is cos(π/3) = 1/2.
With this knowledge, let's create a line. We'll use point-slope form, since we already have a point (π/3, sqrt(3)/2) and the slope (1/2):
(y - sqrt(3)/2) = (1/2) * (x - π/3)
To approximate sin(55°), we can just plug in for x and solve for y:
(y - sqrt(3)/2) = (1/2) * (x - π/3)
y = (1/2) * (11π/36 - 12π/36) + sqrt(3)/2
y = sqrt(3)/2 - π/72
Therefore, our approximation is sin(55°) ≈ sqrt(3)/2 - π/72
If we want to peak at a calculator, we see that sqrt(3)/2 - π/72 ≅ 0.822, and sin(55°) ≈ 0.819, so it seems our approximation isn't half bad.

Doug C.
And here is a Desmos graph to help visualize why using the tangent line at pi/3 gives a good approximation for sin(55pi/180). Notice the very small difference between the y-value of the tangent line and the y-value of the sin(x) at 55pi/180 (radian equivalent to 55 degrees) desmos.com/calculator/0ggtbq96kt10/26/23