Which of the following is an equation of the line tangent to the graph of $y=\cos x$ at $x=\frac{\pi }{2}$?

• A. $y=x-1$
• B. $y=0$
• C. $y=x$
• D. $y=-x+1$

Answer: (B)

Verified step by step guidance

1. Step 1: Recall that the equation of a tangent line to a curve at a given point is given by y = f'(x₀)(x - x₀) + f(x₀), where f'(x₀) is the derivative of the function evaluated at x₀, and f(x₀) is the value of the function at x₀.
2. Step 2: Start by finding the derivative of the function y = cos(x). The derivative of cos(x) is -sin(x), so f'(x) = -sin(x).
3. Step 3: Evaluate the derivative at x = π/2. Substitute x = π/2 into f'(x) = -sin(x) to find the slope of the tangent line. Since sin(π/2) = 1, f'(π/2) = -1.
4. Step 4: Evaluate the function y = cos(x) at x = π/2 to find the y-coordinate of the point of tangency. Since cos(π/2) = 0, the point of tangency is (π/2, 0).
5. Step 5: Use the point-slope form of the equation of a line, y - y₀ = m(x - x₀), where m is the slope and (x₀, y₀) is the point of tangency. Substitute m = -1, x₀ = π/2, and y₀ = 0 into the equation to find the tangent line equation.