Question

Use a linear approximation of f(x) = cos(x) at x = 5π/4 to approximate cos(227°). Give your answer rounded to four decimal places. For example, if you found cos(227°) ~ 0.86612, you would enter 0.8661

Answer 1

To approximate cos(227°) using a linear approximation, find the equation of the tangent line to the function f(x) = cos(x) at x = 5π/4, and substitute x = 227° into it.

Step-by-step explanation:

To find the approximation of cos(227°) using a linear approximation of f(x) = cos(x) at x = 5π/4, we need to find the equation of the tangent line to the curve at x = 5π/4. The equation of a tangent line to a curve can be written as y = f'(a)(x - a) + f(a), where a is the x-coordinate of the point where the tangent line is desired, and f'(a) is the derivative of the function at that point.

Since f(x) = cos(x), the derivative f'(x) = -sin(x). At x = 5π/4, the derivative f'(5π/4) = -sin(5π/4) = -√2/2. Plugging these values into the equation of the tangent line, we get y = (-√2/2)(x - 5π/4) + cos(5π/4).

Now, substitute x = 227° (which is equal to 227π/180 in radians) into the equation of the tangent line, and round the result to four decimal places. This will give you the approximation of cos(227°).

Answer 2

To approximate cos(227°) using a linear approximation of cos(x) at x = 5π/4, we use the linearization formula L(x) = f(a) + f'(a)(x - a). After converting the angle to radians and computing the values, the linearization yields a close approximation of the cosine at that angle.

Step-by-step explanation:

To approximate cos(227°) using a linear approximation of f(x) = cos(x) at x = 5π/4, we need to implement the process of linearization. Linearization approximates the value of a function at a point close to where it is already known.

The formula for the linear approximation of a function f at a point a is given by:

L(x) = f(a) + f'(a)(x - a)

First, we convert the angle from degrees to radians since trigonometric functions in calculus are typically computed in radians.

227° = 227° × (π/180°) ≈ 3.9635 radians

At x = 5π/4, cos(5π/4) = -√2/2, and the derivative of cos(x) is -sin(x), so -sin(5π/4) = -(-√2/2) = √2/2.

The linear approximation L(x) at x = 5π/4 is:

L(x) = -√2/2 + (√2/2)(x - 5π/4)

Plugging in x ≈ 3.9635 radians, we get:

L(3.9635) ≈ -√2/2 + (√2/2)(3.9635 - 5π/4)

When computed and rounded to four decimal places, the result is the approximate value for cos(227°).