Function Point \[ y=\cos (3 x) \quad\left(\frac{\pi}{4},-\frac{\sqrt{2}}{2}\right) \] (a) Find an equation of the tangent line to the graph of the function at the given polnt. (Let \( x \) be the Independent variable and \( y \) be the dependent variable.) \( y=\cos (3 x) \)

We start with the function:   y = cos(3x). Step 1. Find the derivative of y with respect to x to determine the slope of the tangent line.   y′ = d/dx [cos(3x)]. Using the chain rule:   y′ = -sin(3x) · 3 = -3 sin(3x). Step 2. Evaluate the derivative at x = π/4.   y′(π/4) = -3 sin(3(π/4)) = -3 sin(3π/4). Recall that sin(3π/4) = √2/2. Therefore:   y′(π/4) = -3(√2/2) = -(3√2)/2. This is the slope (m) of the tangent line at the point. Step 3. Write the equation of the tangent line using the point-slope form:   y - y₁ = m (x - x₁), where (x₁, y₁) = (π/4, -√2/2) and m = -(3√2)/2. Substitute these values:   y - (-√2/2) = -(3√2)/2 (x - π/4)   y + (√2)/2 = -(3√2)/2 (x - π/4). This is the equation of the tangent line. You can leave the answer in point-slope form or simplify further. For example, solving for y, we have:   y = -(3√2)/2 (x - π/4) - (√2)/2. Thus, the equation of the tangent line is:   y + (√2)/2 = -(3√2)/2 (x - π/4), or equivalently:   y = -(3√2)/2 (x - π/4) - (√2)/2. This completes the solution.