Find the slope of the tangent line to the curve y = sin(x) at x = π/4.

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Question: Find the slope of the tangent line to the curve y = sin(x) at x = π/4.

Options:

Correct Answer: √2/2

Solution:

The derivative f\'(x) = cos(x). At x = π/4, f\'(π/4) = cos(π/4) = √2/2.

Find the slope of the tangent line to the curve y = sin(x) at x = π/4.

Find the slope of the tangent line to the curve y = sin(x) at x = π/4.

Step 1: Identify the function we are working with, which is y = sin(x).
Step 2: Understand that we need to find the slope of the tangent line at a specific point, which is x = π/4.
Step 3: Recall that the slope of the tangent line at a point is given by the derivative of the function.
Step 4: Find the derivative of the function y = sin(x). The derivative is f'(x) = cos(x).
Step 5: Now, we need to evaluate the derivative at the point x = π/4. So we calculate f'(π/4) = cos(π/4).
Step 6: Recall the value of cos(π/4). It is √2/2.
Step 7: Therefore, the slope of the tangent line to the curve y = sin(x) at x = π/4 is √2/2.

Derivative – The derivative of a function at a point gives the slope of the tangent line to the curve at that point.
Trigonometric Functions – Understanding the values of trigonometric functions, particularly sine and cosine, at specific angles.