Calculus question...?

Question

How can I use differentials(or a linear approximation) to estimate the given number: cos31.5 degrees?

Answer 1

You do want to do this in radians, but happily the conversion is easy. I will presume you know how to do that...

You know the cosine of 30 degrees. You can get the slope of the cosine curve at that point. Use that as the slope of a tangent line and find its value 1.5 degrees further on. This will be your approximation to the cosine of 31.5 degrees. Obviously, since you're using the slope of the tangent line, you're making a linear approximation, which will get worse as you move further from the tangent point. However, it is an approximation...

Answer 2

you can use the taylor series of cos(x)=1-x^2/2+ x^4/4!- x^6/6!+ x^8/8! . .


but x has to be in radians.

Answer 3

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

i'll keep it in degrees for now but for small angle approx. you must use radians.

cos(31.5) = cos(30 + 1.5)

cos(30 + 1.5) = cos(30)cos(1.5) - sin(30)sin(1.5)

small angle approximation:
if θ is small then
cos(θ) ≈ 1 sin(θ) ≈ θ where θ is in radians, so...

cos(30) = √3/2
cos(1.5) ≈ 1
sin(30) = 1/2
sin(1.5) ≈ (1.5)π/180

cos(31.5) ≈ √3/2 - (0.5)(0.0262)

cos(31.5) ≈ 0.8529