Use an appropriate local linear approximation to estimate the value of cotangent 44 degrees.
cotan 45 = 1
d cotan/dx = -csc^2 x
d cotan = -csc^2 x dx
here
csc^2 45 = (2/sqrt2)^2 = 4/2 = 2
dx = -1 deg * pi radians/180 deg = -.01745
and
y(x+dx) = y(x) + dy/dx * dx
so
cotan(44)= cotan (45) - 2(-.01745))
= 1 + .035
=1.035
To estimate the value of cotangent 44 degrees using a local linear approximation, we can use the tangent function as a reference. The tangent function is defined as the ratio of the sine function to the cosine function: tan(x) = sin(x)/cos(x).
To approximate cotangent 44 degrees, we first need to find the tangent of an angle close to 44 degrees, for which we know the exact value. Let's take 45 degrees as our reference angle, as it has a well-known tangent value. The tangent of 45 degrees is 1: tan(45) = 1.
Now, we can calculate the slope (or derivative) of the tangent function at 45 degrees. The derivative of the tangent function is known to be sec^2(x). Therefore, the derivative of the tangent function at 45 degrees is sec^2(45) = 1/2.
Let's define a linear function using the formula for a line. We'll use the point-slope form: y - y1 = m(x - x1), where (x1, y1) is the point where we know the exact value, and m is the slope.
Using (x1, y1) = (45, 1) and m = 1/2, the equation becomes: y - 1 = (1/2)(x - 45).
Now, we can substitute x = 44 into the equation to estimate the value of cotangent 44 degrees: y - 1 = (1/2)(44 - 45).
Simplifying the equation, we have: y - 1 = -1/2. Rearranging the equation: y = 1 - 1/2 = 1/2.
Therefore, using the local linear approximation, the estimated value of cotangent 44 degrees is 1/2.