Two curves are said to cross at right angles if their...

Two curves are said to cross at right angles if their tangents are perpendicular at the crossing point. The technical word for "crossing at right angles" is orthogonal. Show that the curves $y=\sin 2 x$ and $y=-\sin (x / 2)$ are orthogonal at the origin. Draw both graphs and both tangents in a square viewing window.

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Key Concepts

Orthogonal Curves

Orthogonal curves are two curves that intersect such that the angle between their tangent lines at the point of intersection is 90 degrees. This concept is important in various fields where the independent behavior or optimal intersections of paths is studied.

Tangent Lines and Perpendicularity

A tangent line to a curve at a point is the line that best approximates the curve near that point. Two curves are orthogonal at their intersection if their tangent lines are perpendicular—that is, if the product of their slopes equals -1.

Differentiation of Functions

Differentiation is the process of calculating a derivative, which represents the instantaneous rate of change or slope of a function at a specific point. This technique is used to determine the tangent line to a curve.

Product of Slopes Criterion

For any two non-vertical lines, the lines are perpendicular if and only if the product of their slopes is -1. This criterion is a key tool in verifying whether two curves intersect at right angles.

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