True or False The slope of the normal line to the curve x=3 cost, y=3 sint at t=π/ 4 is -1 . Jus

step 1 In this problem, we have to determine if a statement is true or false, given our knowledge of pa

True or False The slope of the normal line to the curve x=3...

Key Concepts

Tangent and Normal Lines

The tangent line to a curve at a given point reflects the direction in which the curve is heading at that point, while the normal line is perpendicular to this tangent. The relationship between their slopes is established by the fact that if the tangent line has a slope m, then the slope of the normal line is –1/m, assuming m is not zero. This concept is essential for understanding how different lines interact with curves, particularly in problems involving angles, intersections, and orthogonality.

Evaluation of Trigonometric Functions in Parametric Contexts

In many parametric equations, trigonometric functions such as sine and cosine are used to describe the coordinates. Evaluating these functions at specific parameter values provides the exact point on the curve, and the derivatives at these points yield the necessary information about the slope of the tangent. This approach is crucial when the curve represents common shapes like circles, where trigonometric functions naturally describe the geometry.

Parametric Differentiation

For curves defined by parametric equations, obtaining the slope of the tangent line involves differentiating both the x-coordinate and y-coordinate with respect to the parameter and then taking the ratio of these derivatives (dy/dx). This process is fundamental for analyzing the behavior of curves in the plane and allows one to calculate the instantaneous rate of change at any point along the curve.

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