Use an appropriate local linear approximation to estimate the value of Cotangent 46 degrees.

Use a Taylor series based at 45 degrees, and operate in radians for the derivative and the extra 1 degree.

cot 46 = cot 45 + d(cotx)/dx *(x - pi/4)
= cot 45 + d(cotx)/dx * pi/180
= 1 + (dcotx)/dx*(pi/180)
= 1 - csc^2(pi/4)*pi/180
= 1 - pi/90 = 0.9651

The exact answer is 0.96569

To estimate the value of Cotangent 46 degrees using local linear approximation, we can use the fact that the tangent function can be approximated linearly near certain points.

The tangent function is defined as:

tan(x) = sin(x)/cos(x)

Since the cotangent function is the reciprocal of the tangent function, we can write:

cot(x) = 1/tan(x)

Now, we'll find two angles closest to 46 degrees for which we know the tangent values. Let's use 45 degrees and 50 degrees.

The tangent of 45 degrees is easy to calculate:

tan(45 degrees) = 1

We can use the small-angle approximation to estimate the tangent value at 50 degrees:

tan(50 degrees) ≈ tan(45 degrees) + (50 - 45) * sec^2(45 degrees) * (π/180)

Using the fact that sec(x) = 1/cos(x), we can simplify this expression:

tan(50 degrees) ≈ 1 + (50 - 45) * (1/cos(45 degrees))^2 * (π/180)

Once we have these values, we can calculate the cotangent of 46 degrees as follows:

cot(46 degrees) ≈ 1 / (tan(45 degrees) + (46 - 45) * (1/cos(45 degrees))^2 * (π/180))

cot(46 degrees) ≈ 1 / (1 + (46 - 45) * (1/cos(45 degrees))^2 * (π/180))

Now, substituting the calculated value for cos(45) and evaluating the expression:

cot(46 degrees) ≈ 1 / (1 + (1) * (1/(√2))^2 * (π/180))

cot(46 degrees) ≈ 1 / (1 + (1) * (1/2)^2 * (π/180))

cot(46 degrees) ≈ 1 / (1 + 1/8 * (π/180))

cot(46 degrees) ≈ 1 / (1 + 0.0174533)

cot(46 degrees) ≈ 1 / (1.0174533)

Simplifying this expression, we find:

cot(46 degrees) ≈ 0.9827937

Therefore, the estimated value of cotangent 46 degrees using local linear approximation is approximately 0.9827937.

To estimate the value of the cotangent of 46 degrees using local linear approximation, we need to use the concept of the tangent function and its relation to the cotangent function.

The tangent function is defined as the ratio of the sine function to the cosine function: tan(x) = sin(x) / cos(x).

Since the cotangent function is the reciprocal of the tangent function, we can write it as: cot(x) = 1 / tan(x).

To estimate the value of cot(46°), we can start by finding the tangent of a nearby angle, such as 45°, where the tangent is easily calculated to be 1.

Now, let's consider a small change in angle, Δx, from 45° to 46°. The cotangent function can be approximated by its linear approximation for small angles:

cot(x) ≈ cot(45°) + cot'(45°) * Δx,

where cot'(45°) represents the derivative of cotangent function evaluated at 45°.

To find cot'(45°), we can differentiate the cotangent function with respect to x:

cot'(x) = -csc^2(x),

where csc(x) is the reciprocal of the sine function.

At 45°, the sine function has a value of 1/√2, so we can substitute these values into the formula:

cot'(45°) = -csc^2(45°) = -1/(sin^2(45°)) = -1/(1/√2)^2 = -2.

Now, we can calculate the estimated value of cot(46°):

cot(46°) ≈ cot(45°) + cot'(45°) * Δx
≈ 1 + (-2) * (46 - 45)
≈ 1 - 2
≈ -1.

Therefore, the estimated value of cot(46°) using local linear approximation is approximately -1.