Linear approximation of sin(47) produces wrong answer?

Question

When I attempt to use linear approximation to solve for sin(47) I get a completely incorrect answer...

Here is what I did

sin(47) ~= sin(45) + cos(45)(47-45)

= (sqrt(2) / 2) + ( sqrt(2) / 2 )*(2)

= (3*sqrt(2)) / 2

= 2.12

Which is totally wrong... what am I missing?

Answer 1

not sure what you are doing, but i suspect that in order to have a linear approximation, you need to be working in RADIANS. this will not change the value of sin or cos, but make a huge difference to " 47 - 45"

Answer 2

the derivative of sinx is cosx, if x is measured in radians.

the sine of 45 radians is not sqrt(2)/2

convert 47 degrees to radians, and use the fact that you know the exact value at pi/4 radians.

Answer 3

sin(45 * pi/180) + cos(45 * pi/180) * (47 * pi/180 - 45 * pi/180)
= (1/sqrt(2)) + (1/sqrt(2)) (2 pi/180)
= 1/sqrt(2) + 1/(90*sqrt(2))
= 0.7317894641763

Answer 4

The formula for linear approximation is f '(a)(x-a)+f(a). for the reason that a=3 in this mission: f '(3)(x-3)+f(3) We already keep in mind that f(3)=2, so now all we ought to do is locate f '(x) and replace in 3. we come across the by-product by ability of the chain rule: f '(x)= (a million/3)(3x-a million)^(-2/3)*(3) = (3x-a million)^(-2/3) Now, substituting in x=3, we get... f '(3)=(8)^(-2/3)=a million/4. hence, the linear approximation is (a million/4)(x-3)+2. i imagine you may simplify that.