Final answer:
To find the equation of the tangent to the curve y = cos²(x) at x = π/6, we first differentiate the curve to find the slope of the tangent at that point, and then use the point-slope form of a line to obtain the tangent line's equation.
Step-by-step explanation:
To find the equation of the tangent to the curve y = cos²(x) at x = π/6, we first need to find the derivative of the curve. Let's use the chain rule to differentiate y = cos²(x):
y' = d/dx [cos²(x)] = 2cos(x) * (-sin(x)) = -2cos(x)sin(x).
Now, we evaluate the derivative at x = π/6:
y'(π/6) = -2cos(π/6)sin(π/6) = -2(√3/2)(1/2) = -√3/2.
The slope of the tangent at x = π/6 is -√3/2. To find the equation of the tangent line, use the point-slope form:
y - y1 = m(x - x1),
where m is the slope and (x1, y1) is a point on the tangent. Plugging in the values, we get:
y - cos²(π/6) = -√3/2(x - π/6).
Simplify the point on the curve to obtain cos²(π/6) = (3/4) and write the tangent equation:
y - 3/4 = -√3/2(x - π/6).
This is the equation of the tangent line.